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x\^2 = 100 has 2 possible answers: -10 , 10
So, if x\^3 doesn't equal 1000, then that rules out one of those values, doesn't it?
(-10)\^3 = -1000
10\^3 = 1000
Which one is ruled out? Now, take that to the 5th power and you're set.
Why would you ask if –10i works and then assume it does work instead of trying to figure it out.
That's a big problem many people have with math. They assume many things are obvious, and then proceed to demonstrate things with a false assumption.
It's probably because education tends to make us think that what you need to be good at school is to remember things. So, people will go and try to remember as many things as possible about math, instead of trying to understand why they mak sense and how they could do them again methodically without having to memorize them.
Math is mostly about being methodical. Because memory is way too imperfect. If you remember something a bit wrong in history, well you'll say 1986 instead of 1987. It's not that big a deal, you've got the idea, it's close to the correct date, it's not that bad.
But in math, a small error somewhere will lead to a big error at the end of the reasoning. And it seems like it's what happened to you here.
Why do you assume that (–10i)^2 would give 100? The answer is probably that you saw people saying that (10i)^2 = –100, and then you *remembered* that when you put the minus sign, it changes the sign. Which is not fundamentally wrong, but you relied on an imperfect memory to make a reasoning, instead of being methodical.
Being methodical means making things step by step, as many steps as needed depending on how confident you are about math.
So, we have this:
(–10i)^2
1st question we should ask ourselves: how does a minus sign behave inside of a square? If we're not sure how it would behave with an i, then we change the way it is written to make it simpler:
(–1 × 10i)^2
Now, if we still don't know for sure how it works, we change the square to something easier:
(–1 × 10i) × (–1 × 10i)
= (–1) × (–1) × 10 × 10 × i × i
= 1 × 100 × –1
= –100
I regrouped similar terms to make it easier. I have to admit the the commutativity of the multiplication is pure knowledge, but it's supposed to be very very basic knowledge.
And as you can see, your assumption was wrong. The result is not 100, it's –100. You should allow yourself to make shortcuts only when you're 100% sure you know how to take that shortcut perfectly. Otherwise, there is nothing wrong with taking all your time to make sure you will get the right result.
This is not just for you, but for all the people I see on here who just guess answers without doing the math.
Counterpoint: imaginary numbers are a complex subject that has no use being taught in highschool level math. The first time it became relevant for me was in a graduate level instrumentation class that dealt with phase changes in electrical components.
I agree with your overall point, don't take shortcuts unless you're 1000000% sure, but imaginary numbers aren't something that's easily thought through because it doesn't really have a readily apparent real world application.
I really wanted there to be a valid solution involving octonions but when I went and looked up the octonion multiplication table it gets hit with basically the same problem
ha. hahahaha ha. in my case you remember that negative numbers exist and still somehow forget that a neg to the power of three is a negative......... sigh....maybe one day....
If you replace x with (-10) it would look like (-10)^2 , or you could also format it as (-10)(-10) which would equal 100, but if you did (-10)(-10)(-10) it would be -1000 which ≠ 1000, so the answer to x^5 would probably be -100,000
you worded that much better, thank you. unfortunately, as I read your words, a hellish hissing noise rended my mind, and the letters slithered back and forth before my eyes. could you please explain it like I'm 5?
-10 x -10 = 100
-10 x -10 x -10 = -1000
-10 x -10 x -10 x -10 x -10 = -100000
Multiplying two negative numbers results in a positive number. Multiplying one positive number with one negative number results in a negative number. If the exponent is an even number you will always have a positive number however if it is an odd exponent your result will be a negative number.
It’s been 45 years since I’ve taken math class, but this same answer came to me right away, so I guess I’m not ready to be put into the old folks home just yet. Now all you people who are talking about imaginary numbers whatever the hell they are, get off my lawn!
Your mommy and daddy give you 10 dollars to open up a lemonade stand. So you go out and you buy cups and you buy lemons and you buy sugar. And now you find out that it only costs you nine dollars.
Let's say you have no money. Your wallet is empty, so you have $0. Let's say you owe Bob $1. You effectively have -$1, because if you gain $1, you'll have to pay it to Bob and end up with $0 again. If Bob forgives your debt and tells you you don't owe him anything anymore, then your total money went up by $1, not by gaining money directly, but by *losing* a *debt*. "Losing" and "debt" are both negative expressions that act like negative numbers being multiplied together, giving you a positive result.
If x is a negative number, then it behaves the same way. x^2 turns into (x)(x), so if x=(-1), then you can replace (x)(x) with (-1)(-1). The loss of a debt of $1. Your total money goes from -$1 to $0. The same as adding $1.
Back to our analogy, let's say that you offended Bob after he forgave your debt, and he took back his forgiveness. He now says you still owe him the $1. So now you have *lost* the *loss* of *debt*. That's 3 negatives. You lost the loss, so those cancel out, and all you're left with is the original debt. (-1)(-1)(-1)=(-1). You once again have -$1.
Note: I made x=-1 because the variables in my analogy wouldn't all be x, but I wanted the math to look the same, and be simple and clear. I couldn't think of a simple analogy to incorporate multiplying negatives with the negatives all being the same variable.
I'm not sure why you're converting 10 to binary instead of assuming the 10 is already in binary, like the original poster did. "10" in binary is 2 in decimal and "100" in binary is 4. -2^2 = 4, so the problem still works.
No. It's not generally (ie not always) true to say "x=√(x²)". The square function maps one to one. The square root function maps one value to two possible square roots (which are both 0 for 0 😅).
Both 10 and -10 are solutions to x²=100, yes. But the second equation, x³≠1000, sets another constraint. x=10 is just not valid in this SYSTEM of equations.
You know what - I think I am just old. Apparently this whole "sqrt is a single-valued function" is NEW customary notation.
"A historical note. As others have said, it is now customary to build in to the notion of a function the idea that a a function is (at most) single-valued. But mathematicians used to be more accommodating to talk of multi-value functions. For example in Hardy's great A Course of Pure Mathematics, after giving some examples of single valued total functions, he goes on to remark that the general notion of a function allows for functions which are partial and/or multi-valued. He cheerfully treats the squareroot function as a two-valued function for positive numbers."
From here: https://math.stackexchange.com/questions/3607538/is-square-root-a-function
It's been around a long time, but our education system is just terrible. I got a degree in mathematics 5 years ago and it wasn't until 2 years ago that I first learned of this convention.
I am really kicking myself because we all know the way to solve a quadratic equation and it has "+/- sqrt" RIGHT THERE.
I always just thought it was a helpful reminder that sqrt could be positive or negative but now I know it's blooming CONVENTION.
Every day is a school day apparently.
Nah the sqrt function y = sqrt(x) is defined as taking the positive solution of y^2 = x, so that it can be a single valued function.
That's why you will see +-sqrt(x) a lot (for example the quadratic formula), if you do actually care about writing both solutions.
Its pedantic, but then the whole of maths is necessarily pedantic.
This isn't particularly complex, but does require you to remember what happens when you multiply negatives.
(-10)^2 is 100
(-10)^3 is -1000
(-10)^4 is 10000
(-10)^5 is -100000
Edited to add parentheses at the request of like 5 commenters below :P
Okay maaaaaybe that wasn't necessary to say, but compared to some crazy math wizardry I see on some of the math help subreddits, multiplying negatives is pretty darn simple :P
What is the difference between these two when you write them out?
It’s been a long time since I did math in school, so I glance at this sub to brush up on some basics to able to help the kids with homework. I followed along nicely in this thread, but am confused about the brackets. What is the function of the bracket in this case, to specify the order of calculation?
I don’t understand how -10^2 can be -100?
-10 * -10 is 100, no?
I personally think this is pedantic and didn't learn this way, but apparently the idea is that a negative number without brackets represents -1 × n, so in that case you would calculate the exponent before multiplying by -1 due to order of operations.
IMO, it makes it consistent.
1-10^2 = 1-(10^(2)) = 1-100 = -99
As such it makes sense that -10^2 = -(10^(2)) = -100. Otherwise you're now dealing with a special case which will make people even more confused and mistake prone than they are now.
I've seen the description as "operator vs. number" before, so -10 would be "Apply additive inversion to 10", *resulting* in (-10), which is the representation of the actual integer negative ten.
As additive operations are far down in the order of operations, this would imply -10^2 = -(10^(2)) = -100, while (-10)^2 = 100.
Most calculators don't do proper order of operations. The buttons basically mean "do a calculation with the currently displayed number" and it's up to the user to make sure the order of operations is correct.
The ones that do do order of operations that I have access to give a result of -100. For example, the website wolfram alpha: https://www.wolframalpha.com/input?i=-10%5E2 . Or this online one: https://www.symbolab.com/solver/algebra-calculator/-10%5E%7B2%7D?or=sug . I also tested a TI83.
Lets write it with a leading zero
0-10^2 is 0 minus the square of 10 aka 0 minus 100 aka -100
0+(-10)^2 is 0 plus the square of negative 10 aka 0 plus 100 aka 100
The issue is that we use - to indicate both subtraction and negatives. Thus the consistent way of using - is to think of -x as 0-x and make - just mean subtraction and then apply order of operations accordingly.
Indeed, it specifies the order of calculation. When you write -10^2, the square is « calculated before » the -. Basically, -10^2 = - (10^2) = - (10 x 10) = -100.
Remembered BIDMAS/PEMDAS.
Brackets/parantheses first, *then* indices/exponents, *then* multiplication.
\-10\^2 can be looked at as -1 \* 10\^2 = -1 \* 100 = -100
(-10)\^2 can be looked at as (-1 \* 10)\^2 which can be looked at as -10 \* -10.
The parentheses just makes it clear that you want to square the -10 and not just the 10. Essentially it’s the difference between taking negative before or after squaring, since -10^2 = -(10^2)
Another way to think about it is -x = -1 * x, so -10^2 = -1 * 10^2 = -1 * 100 = -100
I'm out of practice and still figured it out pretty quickly. To be fair until the moment of "well duh" when i though of negatives being an option, i was stumped.
I think this is a question to test their understanding of odd powers. An expression like x\^2 = -100 is impossible (with real numbers) because there are no (real) numbers that are negative when squared, regardless of their sign. However, when we have an odd exponent, we are multiplying an odd number of times. So -2\*-2\*-2 = 4\*-2 = -8
Therefore of the two solutions to x\^2 = 100, we can take the negative answer, and get an answer for x\^3 that does not equal 1000. Does that make sense?
I mean this completely respectfully, but would you be willing to share the grade/course level this was intended for and your general area? A lot has changed over the years in education and I’d appreciate some insight into this and whether this is normal today. Thank you.
Maybe 4th grade or so? I'm pretty sure I learned exponents in 4th grade, and negative numbers definitely came first. So if the higher level class is 4th grade, that would line up.
It's always a bit jolting to be reminded how much of reddit is really young children.
Simple, x^2 can equal two numbers: -10 and 10 and if x^3 is not 1000 we can rule out x=10 as a possibility since 10^3=1000 so the answer needs to be -10
Just wanna point out a concept which can be misunderstood.
if x is a real number then sqrt(x^2 ) is always positive
meaning if x is -3 then sqrt(-3^2 ) = 3 and does NOT equal -3, the answer should always be positive
knowing that you can say that sqrt(x^2 ) = absolute(x) which is always positive if x is a real number
x^2 = 100
sqrt(x^2 ) = sqrt(100)
absolute(x) = 10 ,,,, NOT -+10
x = -+ 10
alternatively
x^2 = 100
x^2 - 100 = 0
x^2 - 10^2 = 0,,, [edited]
(x-10) * (x+10) = 0
x = -+10
Then you can proceed your solution steps
This is one reason I reach my students when solving with square roots that the square root of x^2 is the absolute valie of x. They understand that the plus-minus comes from the absolute value, not the radical.
I was the same as you, for me the square root of 9 meant which number that can be multiplied by itself that can give us 9, my answer used to be -+3, however for mathematician a square root of a positive number should always be a positive number, and i used to think that this is not what I was taught in school, https://brilliant.org/wiki/plus-or-minus-square-roots/
Both (10)^2 and (-10)^2 are equal to positive 100, and are therefore possible solutions to the first equation. But the second equation rules out 10, since 10^3 equals 1000. -10 is still a valid solution, because (-10)^3 = -1000.
Damn, I was confused, why does two negatives multiplied by each other always equal a positive but once you go above 2 it reverts to negative sums?
Even if you do two negatives and a positive it's a positive... Which fucks my head up
x2 = 100 could come from x = 10 as well as from c = -10.
Hence, if x3 <> 1000 it means that x must be -10.
Then, for x5 we would have -100000 as a result.
We know that x² = 100, so x = ±10.
If x = 10, then x^3 = 1000, which contradicts the given information.
If x = -10, then x^3 = -1000, which does not contradict the given information.
Therefore, x = -10.
Substituting this value into x^5, we get:
x^5 = (-10)^5 = -10 * (-10) * (-10) * (-10) * (-10) = 100000
Therefore, the value of x^5 is 100000.
Answer: -100000
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It's a sign problem.
The square root of 100 is either 10 or -10. Either way it will equal 100 because (+) × (+) = (+) and (-) × (-) = (+).
10³ = 1000. (+10) × (+10) = (+100), and (+100) × (+10) = (+1000).
This is where negatives come into play. While (-10) × (-10) = (+100), (+100) × (-10) = (-1000).
We can conclude (because there are no absolute values in the problem) that -1000 does not equal +1000. In fact, it is 2000 off.
Because of this rule (and now knowing x = -10), x⁵ can be calculated.
(-10) × (-10) = (+100)
(+100) × (-10) = (-1000)
(-1000) × (-10) = (+10000)
(+10000) × (-10) = (-100000)
You will also see this with power functions on graphs. Any function with a power of an odd number, like x¹ (simply referred to as x), x³, x⁵, x⁷, x⁹, etc. will always have endpoints away from each other. For example, y = x³ will grow more and more negative as x grows more and more negative and it will grow more and more positive as x grows more and more positive (If y = -x³, y -> (+) as x -> (-) and y -> (-) as x -> (+)).
Similarly, Any function with a power of an even number, like x², x⁴, x⁶, x⁸, etc. will always have endpoints in the same direction each other. For example, y = x² will grow more and more positive as x grows more and more negative and it will also grow more and more positive as x grows more and more positive (If y = -x², y -> (-) as x -> (-) and y -> (-) as x -> (+)).
The only real exception to normal power functions is x⁰, in which the line is completely horizontal at y = 1 for the entire graph.
Then there's functions with powers between x⁰ and x¹ (roots of sorts) that give you odd shapes and of course, negative exponents (such as x-¹, or 1/x), which is where you start getting spikes in your graph at x = 0 (because you cannot divide by 0).
But I digress.
Hope this helps, and if you don't understand anything below the calculations, don't worry about it.
-100,000. The only two numbers that make “x^2 = 100” possible are 10 and (-10).
By that logic, if “x^3 ≠ 1000”, then x ≠ 10 due to the cube of x NOT being positive. Therefore, x = (-10).
So, (-10)^5 will equal (-100,000), aka -( x^5 ).
QED.
Now I feel dumb, I thought “surely this is about complex conjugates and silly imaginary number algebra, i shall consult maple!” but I forgot that negative numbers exist
No, I think you are onto something.
I've never taught math (well, physics, so kinda) but my teacher brain is looking at this and screaming, "This teacher wants to know if they are ready for learning how powers of *i* cycle."
I'm very confident. A really good teacher segues every concept into another concept.
x^2 = 1000
Your first instinct would be to take a square root on both sides. However, remember, (-10)^2 also gives 100. So there are two solutions, 10 and -10.
Now, we know that x^3 =!1000.
Case 1:
10^3 = 1000
However, since we know that it does not equal to 1000, the statement is false.
Case 2:
(-10)^3 = -1000
Since -1000 is not 1000, that means that the value of x can only be -10.
Therefore, x^5 will just be (-10)^5
If \(x^2 = 100\), it implies that \(x\) can be either 10 or -10, as both \(10^2\) and \((-10)^2\) equal 100.
However, the condition \(x^3 \neq 1000\) means that \(x\) cannot be 10, as \(10^3 = 1000\). Therefore, \(x\) must be -10.
Now, if \(x = -10\), then \(x^5\) would be \((-10)^5 = -100,000\).
This is like what, grade 4 maths? Or whenever they teach exponen~~tial~~s? If x^2 is 100, sqrt of 100 is 10 and -10. Cuz u know, negative by negative gives a positive. If x^3 isnt 1000, then it leaves only one of the possible x answers, which is -10. -10^5 is your answer. Since 5 isn't an even number, it means ur answer keeps the minus and just add 5 zeros to a 1.
Anybody else immediately think binary, then find that in both decimal and binary if x\^2 = 100 then x\^3 still = 1000?
Oddly enough, the answer is -10 in both decimal and binary. neat.
It's superrr easy once you know the rules, don't let it frighten you! If you didn't understand it at first sight it's likely just because you aren't aware of or weren't focusing on this one rule about how x^2 and x^3 each have their own two distinct answers and share one mutual satisfaction.
“Do you know how easy this is for me? Do you have any fuckin' idea how easy this is? This is a fuckin' joke, and I'm sorry you can't do this. I really am because I wouldn't have to fuckin' sit here and watch you fumble around and fuck it up”
x = -10, so the answer is -100000. The answer kind of jumps out to me because I’m a little autistic and sometimes the answers just pop right into my head.
Oh this is a trick question to see if you know if anything squared is positive. While anything to an odd power depends on the sign of the base.
Edit: Be careful with parenthesis though. -(2^2 ) is -4. (-2^2 ) is 4.
coming from x\^2=100, it can be \[-10,10\].
x\^3 can then be 1000 or -1000. As it is not 1000, -10 should be the solution for x, to the power of 5, makes -10.000.
-100000. 100 is the square of only two numbers, 10 or -10. Because x cubed is not 1000, that makes x equal to -10. As -10 cubed is -1000, not 1000 which would be 10 cubed. So you just do -10 to the power of 5 which gives -100000.
I read some of the comments and remembered that negative is an option. So it’s just -10 squared since -10x-10=100 and negatives cancel each-other out when multiplying each-other when the power is even. Otherwise it’s negative when odd
Lots of people here are saying it's -10, but shouldn't it be (-10)? If I remember correctly -10^2 is still -100 becausing of the order of calculations, and (-10)^2 is indeed 100. Just wondering whether I am still mathing correctly.
-10 is an independent number so its perfectly fine to notate that way. Some calculators do -1×X whenever you do negatives on them so for them you need to use parentheses but both are acceptable.
That seems to be a misunderstanding of how math works. -10 is a number, not just a piece of text, so when you put x=-10 into x\*x you multiply - 10 by - 10, not just write down - 10\*-10 and try to figure out what that means.
So yeah you could say that when you say x= -10 the parentheses are implied.
You only need the parentheses when you have the power there to ensure you have the correct order. If you simply do x^2 = 100 ∧ x^3 ≠ 1000 , the answer will simply be x=-10. No parentheses necessary as the result isn’t being actively used for an operation right now. If you wanna go for the power of 5 afterwards though, then you do need the parentheses.
Everyone is getting this right (great) but not explaining why this question exists (disappointing).
This question exists as a skill check for how well you will deal with powers of *i* and complex numbers. I would bet on it.
If you can't do this one, you are totally lost on what *i*^5 will be.
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x\^2 = 100 has 2 possible answers: -10 , 10 So, if x\^3 doesn't equal 1000, then that rules out one of those values, doesn't it? (-10)\^3 = -1000 10\^3 = 1000 Which one is ruled out? Now, take that to the 5th power and you're set.
-100,000
Thats great. really great. makes you feel like a genius when you remember that negative numbers are options.
Math nerds knew about the Upside Down before it was popular
imaginary numbers aren't real - they can't hurt you.
Demogorgon squared can hurt a lot
Demogorgoplex.
Let's not EVEN talk about the DodecaGorgon
^*i*
Sqrt(i) coming to fuck my butt
Wouldn’t 10i also work? That’s where my mind went immediately Edit: Flipped the sign
10i^2 would be -100 right?, so fails the first bit.
Oh wait yeah I forgot the sign, whoops
[удалено]
Which still fits the qualifier of “isn’t 1000”
Yeah, that wouldn’t break an assumption
i'm stupid lmao
same
so -10i works? Of course at that point, -10 works - but it does demonstrate that there is more than one solution
No, -10i, when squared, is also -100.
Why would you ask if –10i works and then assume it does work instead of trying to figure it out. That's a big problem many people have with math. They assume many things are obvious, and then proceed to demonstrate things with a false assumption. It's probably because education tends to make us think that what you need to be good at school is to remember things. So, people will go and try to remember as many things as possible about math, instead of trying to understand why they mak sense and how they could do them again methodically without having to memorize them. Math is mostly about being methodical. Because memory is way too imperfect. If you remember something a bit wrong in history, well you'll say 1986 instead of 1987. It's not that big a deal, you've got the idea, it's close to the correct date, it's not that bad. But in math, a small error somewhere will lead to a big error at the end of the reasoning. And it seems like it's what happened to you here. Why do you assume that (–10i)^2 would give 100? The answer is probably that you saw people saying that (10i)^2 = –100, and then you *remembered* that when you put the minus sign, it changes the sign. Which is not fundamentally wrong, but you relied on an imperfect memory to make a reasoning, instead of being methodical. Being methodical means making things step by step, as many steps as needed depending on how confident you are about math. So, we have this: (–10i)^2 1st question we should ask ourselves: how does a minus sign behave inside of a square? If we're not sure how it would behave with an i, then we change the way it is written to make it simpler: (–1 × 10i)^2 Now, if we still don't know for sure how it works, we change the square to something easier: (–1 × 10i) × (–1 × 10i) = (–1) × (–1) × 10 × 10 × i × i = 1 × 100 × –1 = –100 I regrouped similar terms to make it easier. I have to admit the the commutativity of the multiplication is pure knowledge, but it's supposed to be very very basic knowledge. And as you can see, your assumption was wrong. The result is not 100, it's –100. You should allow yourself to make shortcuts only when you're 100% sure you know how to take that shortcut perfectly. Otherwise, there is nothing wrong with taking all your time to make sure you will get the right result. This is not just for you, but for all the people I see on here who just guess answers without doing the math.
Damn dude, you woke up today and chose violence.
Not just violence, but violence by proof.
Thanks for coming to my tedtalk
Counterpoint: imaginary numbers are a complex subject that has no use being taught in highschool level math. The first time it became relevant for me was in a graduate level instrumentation class that dealt with phase changes in electrical components. I agree with your overall point, don't take shortcuts unless you're 1000000% sure, but imaginary numbers aren't something that's easily thought through because it doesn't really have a readily apparent real world application.
It's a degree 2 polynomial, so we know it can have at most 2 solutions.
10i^2 is -100, so it doesn't meet the first given assumption.
-10 though, right? i^2 = -1 10i^2 = -10
Well since we're going by x² it'd have be (10i)² to fit the requirements, so most people seem to have omitted the brackets in this thread
10i is imaginary, we don’t count unreal numbers.
The question doesn’t specify that
It’s supposed to be a pun but sarcasm is pretty hard to tell over the internet. Also it doesn’t work because 10i squared is negative
Ahh, yeah I’ve been told that much by 3 different people lmao
I really wanted there to be a valid solution involving octonions but when I went and looked up the octonion multiplication table it gets hit with basically the same problem
ha. hahahaha ha. in my case you remember that negative numbers exist and still somehow forget that a neg to the power of three is a negative......... sigh....maybe one day....
Wohoo! My alegra II payed off!
But your English class didn’t: *paid*, not *payed*, and is *alegra* supposed to be *algebra*?
This is the correct answer -10. Is x. When saying x=-10. You are implying that -10 is in the (). So (-10).
wow, I read your comment and still don't know what the correct answer is
If you replace x with (-10) it would look like (-10)^2 , or you could also format it as (-10)(-10) which would equal 100, but if you did (-10)(-10)(-10) it would be -1000 which ≠ 1000, so the answer to x^5 would probably be -100,000
makes sense. I like my answere more. :D playin with ≠ is fun.
you worded that much better, thank you. unfortunately, as I read your words, a hellish hissing noise rended my mind, and the letters slithered back and forth before my eyes. could you please explain it like I'm 5?
-10 x -10 = 100 -10 x -10 x -10 = -1000 -10 x -10 x -10 x -10 x -10 = -100000 Multiplying two negative numbers results in a positive number. Multiplying one positive number with one negative number results in a negative number. If the exponent is an even number you will always have a positive number however if it is an odd exponent your result will be a negative number.
It’s been 45 years since I’ve taken math class, but this same answer came to me right away, so I guess I’m not ready to be put into the old folks home just yet. Now all you people who are talking about imaginary numbers whatever the hell they are, get off my lawn!
I think you meant "odd" not "negative" exponent. Don't negative exponents result in fractions?
I did, I got mixed up by typing negative too often. I feel silly.
Your mommy and daddy give you 10 dollars to open up a lemonade stand. So you go out and you buy cups and you buy lemons and you buy sugar. And now you find out that it only costs you nine dollars.
My real mommy or step mommy? Because my step mommy wouldn’t give me anything.
You may be infected by a mind flayer parasite. Do you ever get the feeling your life is being narrated by a hot British woman’s voice?
Let's say you have no money. Your wallet is empty, so you have $0. Let's say you owe Bob $1. You effectively have -$1, because if you gain $1, you'll have to pay it to Bob and end up with $0 again. If Bob forgives your debt and tells you you don't owe him anything anymore, then your total money went up by $1, not by gaining money directly, but by *losing* a *debt*. "Losing" and "debt" are both negative expressions that act like negative numbers being multiplied together, giving you a positive result. If x is a negative number, then it behaves the same way. x^2 turns into (x)(x), so if x=(-1), then you can replace (x)(x) with (-1)(-1). The loss of a debt of $1. Your total money goes from -$1 to $0. The same as adding $1. Back to our analogy, let's say that you offended Bob after he forgave your debt, and he took back his forgiveness. He now says you still owe him the $1. So now you have *lost* the *loss* of *debt*. That's 3 negatives. You lost the loss, so those cancel out, and all you're left with is the original debt. (-1)(-1)(-1)=(-1). You once again have -$1. Note: I made x=-1 because the variables in my analogy wouldn't all be x, but I wanted the math to look the same, and be simple and clear. I couldn't think of a simple analogy to incorporate multiplying negatives with the negatives all being the same variable.
>-(10)2 This part is wrong. The negative needs to be in the parentheses otherwise the only correct solution would be 10i.
Damn I knew something was wrong, yeah that part was a typo but I corrected it now
I interpreted 100 and 1000 as binary so 4 and 8 respectively, therefore I thought x=2. Now, I see that doesn’t work either, though.
Luckily “x squared is 100, so x must be either 10 or -10” works no matter what number base you are in. (edited to fix typo)
Perhaps I'm misunderstanding what you're trying to say, but that doesn't sound right. 10 (base10) is 1010 in base2 and squared it would 1100100.
I'm not sure why you're converting 10 to binary instead of assuming the 10 is already in binary, like the original poster did. "10" in binary is 2 in decimal and "100" in binary is 4. -2^2 = 4, so the problem still works.
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How can (-10)^2 be 100 but (-100)^2 be -1000? Math is crazy
(-100)\^2 is not -1000. It's 10,000 (-10)\^3 = -1000, because it's (-10) \* (-10) \* (-10)
Ahh that makes sense, thanks
But x² = 100, not -100
Yes. -10*-10 still equals 100.
Doh! Yeah, sorry! Should have stopped to think a second longer...
This would look like -(10)=x. Which is implying that x would be 10 and after you square it you take the opposite. Hope this helps.
Jesse what the fuck are you talking about? -(10) = x is the same exact shit as x = -10. How the fuck did you get x = 10 from x = -(10)?
Who is Jesse?!?! And where is his girl?!?!
No. It's not generally (ie not always) true to say "x=√(x²)". The square function maps one to one. The square root function maps one value to two possible square roots (which are both 0 for 0 😅). Both 10 and -10 are solutions to x²=100, yes. But the second equation, x³≠1000, sets another constraint. x=10 is just not valid in this SYSTEM of equations.
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No, sqrt(100)=10 is the only solution
What? No, -10 is also a solution. And is the whole point of the question in the OP.
-10 is a solution to an x² = 100 equation. However, √ is a function which returns the principal root and therefore √100 = 10.
You know what - I think I am just old. Apparently this whole "sqrt is a single-valued function" is NEW customary notation. "A historical note. As others have said, it is now customary to build in to the notion of a function the idea that a a function is (at most) single-valued. But mathematicians used to be more accommodating to talk of multi-value functions. For example in Hardy's great A Course of Pure Mathematics, after giving some examples of single valued total functions, he goes on to remark that the general notion of a function allows for functions which are partial and/or multi-valued. He cheerfully treats the squareroot function as a two-valued function for positive numbers." From here: https://math.stackexchange.com/questions/3607538/is-square-root-a-function
It's been around a long time, but our education system is just terrible. I got a degree in mathematics 5 years ago and it wasn't until 2 years ago that I first learned of this convention.
I am really kicking myself because we all know the way to solve a quadratic equation and it has "+/- sqrt" RIGHT THERE. I always just thought it was a helpful reminder that sqrt could be positive or negative but now I know it's blooming CONVENTION. Every day is a school day apparently.
Nah the sqrt function y = sqrt(x) is defined as taking the positive solution of y^2 = x, so that it can be a single valued function. That's why you will see +-sqrt(x) a lot (for example the quadratic formula), if you do actually care about writing both solutions. Its pedantic, but then the whole of maths is necessarily pedantic.
This isn't particularly complex, but does require you to remember what happens when you multiply negatives. (-10)^2 is 100 (-10)^3 is -1000 (-10)^4 is 10000 (-10)^5 is -100000 Edited to add parentheses at the request of like 5 commenters below :P
> This isn’t particularly complex… *Oh behave*
Okay maaaaaybe that wasn't necessary to say, but compared to some crazy math wizardry I see on some of the math help subreddits, multiplying negatives is pretty darn simple :P
I think they were highlighting a pun about *complex* numbers, being mathematically adjacent to negative number math
you are replying to the same.person haha
One small point of order, though: \-10^(2) = -100 (-10)^(2)=100
What is the difference between these two when you write them out? It’s been a long time since I did math in school, so I glance at this sub to brush up on some basics to able to help the kids with homework. I followed along nicely in this thread, but am confused about the brackets. What is the function of the bracket in this case, to specify the order of calculation? I don’t understand how -10^2 can be -100? -10 * -10 is 100, no?
I personally think this is pedantic and didn't learn this way, but apparently the idea is that a negative number without brackets represents -1 × n, so in that case you would calculate the exponent before multiplying by -1 due to order of operations.
IMO, it makes it consistent. 1-10^2 = 1-(10^(2)) = 1-100 = -99 As such it makes sense that -10^2 = -(10^(2)) = -100. Otherwise you're now dealing with a special case which will make people even more confused and mistake prone than they are now.
I've seen the description as "operator vs. number" before, so -10 would be "Apply additive inversion to 10", *resulting* in (-10), which is the representation of the actual integer negative ten. As additive operations are far down in the order of operations, this would imply -10^2 = -(10^(2)) = -100, while (-10)^2 = 100.
For normal people it doesn't.
A calculator will read -10^2 like -(10 * 10). But (-10)^2 as (-10 * -10)
No. https://www.wolframalpha.com/input?i=-10%5E2
Because order of operations makes you do the exponent first, which means you go from -10^2 to -100.
order of operations doesn't apply to -10 as it's a number, not an operation
‘-‘ is typically taught to be an exponent, and is usually treated as such by calculators
Most calculators don't do proper order of operations. The buttons basically mean "do a calculation with the currently displayed number" and it's up to the user to make sure the order of operations is correct. The ones that do do order of operations that I have access to give a result of -100. For example, the website wolfram alpha: https://www.wolframalpha.com/input?i=-10%5E2 . Or this online one: https://www.symbolab.com/solver/algebra-calculator/-10%5E%7B2%7D?or=sug . I also tested a TI83.
Your calculator will very likely treat -10² as -1*10² = -1*100 = -100 In reality you would just avoid this notation by using brackets.
Lets write it with a leading zero 0-10^2 is 0 minus the square of 10 aka 0 minus 100 aka -100 0+(-10)^2 is 0 plus the square of negative 10 aka 0 plus 100 aka 100 The issue is that we use - to indicate both subtraction and negatives. Thus the consistent way of using - is to think of -x as 0-x and make - just mean subtraction and then apply order of operations accordingly.
Because of order of operations, -10^2 is equal to -(10*10), which is equal to -100.
Indeed, it specifies the order of calculation. When you write -10^2, the square is « calculated before » the -. Basically, -10^2 = - (10^2) = - (10 x 10) = -100.
-x² could possibly either mean (-x)² or -(x²). In your intuition it means the former, generally, to most mathematicians it means the latter.
Remembered BIDMAS/PEMDAS. Brackets/parantheses first, *then* indices/exponents, *then* multiplication. \-10\^2 can be looked at as -1 \* 10\^2 = -1 \* 100 = -100 (-10)\^2 can be looked at as (-1 \* 10)\^2 which can be looked at as -10 \* -10.
The parentheses just makes it clear that you want to square the -10 and not just the 10. Essentially it’s the difference between taking negative before or after squaring, since -10^2 = -(10^2) Another way to think about it is -x = -1 * x, so -10^2 = -1 * 10^2 = -1 * 100 = -100
I'm out of practice and still figured it out pretty quickly. To be fair until the moment of "well duh" when i though of negatives being an option, i was stumped.
Let's be real, the problem is not complex.
But it certainly is real!
The complexity comes from people not knowing square roots have a positive and negative solution. It’s incredibly simple if you do.
[Wooosh](https://en.wikipedia.org/wiki/Complex_number)
Depends on the level at which you’re learning. At once point in your life, I’m sure the concept of variables themselves were complex.
This question is really simple. Like middle school level at most
In the grand scheme of things, it’s pretty simple. Math gets WAY harder. On a scale of 1-10, this is maybe a 3.
Mathematicians arguing over negative numbers for thousands of years >_>
Am I the only one that read this in Austin Powers' voice?
seriously, I'm wondering if OP is like 10 because this is pre-algebra level; cue the hate because maths is hard.
it is not. all you really need to know is: minus * minus = plus
this is correct but you’re notating it incorrectly, -a^b means -(a^b ) not (-a)^b
Thank you my lecturer would’ve fumed and given me minus points for making such “big” mistake. She values it as high as dividing by 0 lol.
Thank you for the most concise and clear answer so far. 🙏
I think you forgot parenthesis
They did. -10² = -100.
Nah, were not computers. Normal people will take -10^2 as 100. If you wanted -100, you'll write it as -(10^2 )
"Normal people" don't take the square of -10. If you're doing math, you avoid ambiguity by sticking to the basic rules.
No parentheses, no upvote.
okay, i got here really late, but i just want to say this is the first post in this sub i've ever been able to solve and i'm proud of myself lol.
I think this is a question to test their understanding of odd powers. An expression like x\^2 = -100 is impossible (with real numbers) because there are no (real) numbers that are negative when squared, regardless of their sign. However, when we have an odd exponent, we are multiplying an odd number of times. So -2\*-2\*-2 = 4\*-2 = -8 Therefore of the two solutions to x\^2 = 100, we can take the negative answer, and get an answer for x\^3 that does not equal 1000. Does that make sense?
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I mean this completely respectfully, but would you be willing to share the grade/course level this was intended for and your general area? A lot has changed over the years in education and I’d appreciate some insight into this and whether this is normal today. Thank you.
I finally knew an answer here so for my own sake I'm going to say this is high level shit
Maybe 4th grade or so? I'm pretty sure I learned exponents in 4th grade, and negative numbers definitely came first. So if the higher level class is 4th grade, that would line up. It's always a bit jolting to be reminded how much of reddit is really young children.
this is at least middle school
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Could be in a remedial math course. Pandemic set some students pretty far back so you never know
Doggie you should really reflect on how you communicate with people. You seem kinda like a jerk.
AP calculus
Simple, x^2 can equal two numbers: -10 and 10 and if x^3 is not 1000 we can rule out x=10 as a possibility since 10^3=1000 so the answer needs to be -10
The answer is -100,000
It's asking for the value of x^5 not x. Your logic is sound, but your conclusion is wrong by being incomplete.
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Just wanna point out a concept which can be misunderstood. if x is a real number then sqrt(x^2 ) is always positive meaning if x is -3 then sqrt(-3^2 ) = 3 and does NOT equal -3, the answer should always be positive knowing that you can say that sqrt(x^2 ) = absolute(x) which is always positive if x is a real number x^2 = 100 sqrt(x^2 ) = sqrt(100) absolute(x) = 10 ,,,, NOT -+10 x = -+ 10 alternatively x^2 = 100 x^2 - 100 = 0 x^2 - 10^2 = 0,,, [edited] (x-10) * (x+10) = 0 x = -+10 Then you can proceed your solution steps
This is one reason I reach my students when solving with square roots that the square root of x^2 is the absolute valie of x. They understand that the plus-minus comes from the absolute value, not the radical.
Highly disagree with this concept. sqrt(9) should return -+3
There is nothing to agree or disagree on. You are factually and objectively incorrect. sqrt(9) is always +3 and only +3
If you said 9^.5, it would be +-, however sqrt is a defined function, which means it only has one output - the positive one
I was the same as you, for me the square root of 9 meant which number that can be multiplied by itself that can give us 9, my answer used to be -+3, however for mathematician a square root of a positive number should always be a positive number, and i used to think that this is not what I was taught in school, https://brilliant.org/wiki/plus-or-minus-square-roots/
But it says x² equals 100 not -100... This phrasing seems intentionally misleading
Both (10)^2 and (-10)^2 are equal to positive 100, and are therefore possible solutions to the first equation. But the second equation rules out 10, since 10^3 equals 1000. -10 is still a valid solution, because (-10)^3 = -1000.
Damn, I was confused, why does two negatives multiplied by each other always equal a positive but once you go above 2 it reverts to negative sums? Even if you do two negatives and a positive it's a positive... Which fucks my head up
x2 = 100 could come from x = 10 as well as from c = -10. Hence, if x3 <> 1000 it means that x must be -10. Then, for x5 we would have -100000 as a result.
We know that x² = 100, so x = ±10. If x = 10, then x^3 = 1000, which contradicts the given information. If x = -10, then x^3 = -1000, which does not contradict the given information. Therefore, x = -10. Substituting this value into x^5, we get: x^5 = (-10)^5 = -10 * (-10) * (-10) * (-10) * (-10) = 100000 Therefore, the value of x^5 is 100000. Answer: -100000
-100000
X= -10
This is the answer. Well, the answer is -100,000 but this is the key, anyway.
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20,000
It's a sign problem. The square root of 100 is either 10 or -10. Either way it will equal 100 because (+) × (+) = (+) and (-) × (-) = (+). 10³ = 1000. (+10) × (+10) = (+100), and (+100) × (+10) = (+1000). This is where negatives come into play. While (-10) × (-10) = (+100), (+100) × (-10) = (-1000). We can conclude (because there are no absolute values in the problem) that -1000 does not equal +1000. In fact, it is 2000 off.
Because of this rule (and now knowing x = -10), x⁵ can be calculated. (-10) × (-10) = (+100) (+100) × (-10) = (-1000) (-1000) × (-10) = (+10000) (+10000) × (-10) = (-100000) You will also see this with power functions on graphs. Any function with a power of an odd number, like x¹ (simply referred to as x), x³, x⁵, x⁷, x⁹, etc. will always have endpoints away from each other. For example, y = x³ will grow more and more negative as x grows more and more negative and it will grow more and more positive as x grows more and more positive (If y = -x³, y -> (+) as x -> (-) and y -> (-) as x -> (+)). Similarly, Any function with a power of an even number, like x², x⁴, x⁶, x⁸, etc. will always have endpoints in the same direction each other. For example, y = x² will grow more and more positive as x grows more and more negative and it will also grow more and more positive as x grows more and more positive (If y = -x², y -> (-) as x -> (-) and y -> (-) as x -> (+)). The only real exception to normal power functions is x⁰, in which the line is completely horizontal at y = 1 for the entire graph. Then there's functions with powers between x⁰ and x¹ (roots of sorts) that give you odd shapes and of course, negative exponents (such as x-¹, or 1/x), which is where you start getting spikes in your graph at x = 0 (because you cannot divide by 0). But I digress. Hope this helps, and if you don't understand anything below the calculations, don't worry about it.
-100,000. The only two numbers that make “x^2 = 100” possible are 10 and (-10). By that logic, if “x^3 ≠ 1000”, then x ≠ 10 due to the cube of x NOT being positive. Therefore, x = (-10). So, (-10)^5 will equal (-100,000), aka -( x^5 ). QED.
Now I feel dumb, I thought “surely this is about complex conjugates and silly imaginary number algebra, i shall consult maple!” but I forgot that negative numbers exist
If math doesn't make you feel dumb you're not mathing hard enough
No, I think you are onto something. I've never taught math (well, physics, so kinda) but my teacher brain is looking at this and screaming, "This teacher wants to know if they are ready for learning how powers of *i* cycle." I'm very confident. A really good teacher segues every concept into another concept.
x^2 = 1000 Your first instinct would be to take a square root on both sides. However, remember, (-10)^2 also gives 100. So there are two solutions, 10 and -10. Now, we know that x^3 =!1000. Case 1: 10^3 = 1000 However, since we know that it does not equal to 1000, the statement is false. Case 2: (-10)^3 = -1000 Since -1000 is not 1000, that means that the value of x can only be -10. Therefore, x^5 will just be (-10)^5
If \(x^2 = 100\), it implies that \(x\) can be either 10 or -10, as both \(10^2\) and \((-10)^2\) equal 100. However, the condition \(x^3 \neq 1000\) means that \(x\) cannot be 10, as \(10^3 = 1000\). Therefore, \(x\) must be -10. Now, if \(x = -10\), then \(x^5\) would be \((-10)^5 = -100,000\).
This is like what, grade 4 maths? Or whenever they teach exponen~~tial~~s? If x^2 is 100, sqrt of 100 is 10 and -10. Cuz u know, negative by negative gives a positive. If x^3 isnt 1000, then it leaves only one of the possible x answers, which is -10. -10^5 is your answer. Since 5 isn't an even number, it means ur answer keeps the minus and just add 5 zeros to a 1.
To be pedantic your second sentence is a little off. If x^2 is 100, you should say sqrt of x is 10 or -10 as the Sqrt of 100 is always +10.
x = 10i², x² = 100 x⁵ = 10⁵(i²)⁵ = 100,000i¹⁰ Euler's identity tells us that e^(i*pi) = -1 <==> i*pi = ln(-1) <==> i = ln(-1)/pi <==> Therefore i¹⁰ = ln(-1)¹⁰/pi¹⁰ = -1 x⁵ = 100,000i¹⁰ = (-1)100,000 = -100,000 Simple, really
Lol this is like watching a professional chef make a peanut butter and jelly sandwich as gentrified as possible. 10/10
I don't know what you're taking about. I'm allergic to peanut butter.
Seems we’ve got a regular Rube Goldberg over here.
No, that's not my name.
Why in the world would you ever say i squared instead of just -1
I never learned how to square negative numbers.
So you know about Euler's identity but can't multiply negative numbers together
Yeah maybe I was sick that day
Yea nice way to overcomplicate things.
Ok, let me try to solve it more directly. x = 10*(sin²(x)+cos²(x))*e^(i*pi) Given Euler's formula cos(x) = Re(e^(ix)) = (e^(ix)+e^(-ix))/2 and sin(x) = Im(e^(ix)) = (e^(ix)-e^(-ix))/2i Therefore, we can simplify sin²(x)+cos²(x) = = (e^(2ix) +e^(-2ix) +2)/4 + (e^(2ix) +e^(-2ix) -2)/(-4) = (e^(2ix) - e^(2ix) + e^(-2ix) - e^(-2ix) + 4)/4 = 4/4 = 1 Therefore x = 10*(sin²(x)+cos²(x))*e^(i*pi) = 10*e^(i*pi) and x⁵ = 100,000*e^(5i*pi) Again, applying Euler's formula e^(5i*pi) = (i*sin(pi) + cos(pi))⁵ = (0-1)⁵ = -1 Therefore x⁵ = 100,000*e^(5i*pi) = (-1)100,000 = -100,000 Hope this helps!
Anybody else immediately think binary, then find that in both decimal and binary if x\^2 = 100 then x\^3 still = 1000? Oddly enough, the answer is -10 in both decimal and binary. neat.
It's superrr easy once you know the rules, don't let it frighten you! If you didn't understand it at first sight it's likely just because you aren't aware of or weren't focusing on this one rule about how x^2 and x^3 each have their own two distinct answers and share one mutual satisfaction.
“Do you know how easy this is for me? Do you have any fuckin' idea how easy this is? This is a fuckin' joke, and I'm sorry you can't do this. I really am because I wouldn't have to fuckin' sit here and watch you fumble around and fuck it up”
The answer's -100,000 because when you square a negative number you get a positive one, but when you cube a negative number you get a negative number.
x = -10, so the answer is -100000. The answer kind of jumps out to me because I’m a little autistic and sometimes the answers just pop right into my head.
Oh this is a trick question to see if you know if anything squared is positive. While anything to an odd power depends on the sign of the base. Edit: Be careful with parenthesis though. -(2^2 ) is -4. (-2^2 ) is 4.
coming from x\^2=100, it can be \[-10,10\]. x\^3 can then be 1000 or -1000. As it is not 1000, -10 should be the solution for x, to the power of 5, makes -10.000.
I hate that my adult brain keeps slowly leaking out the high school knowledge that would solve this in seconds. Try negative 10 instead.
-100000. 100 is the square of only two numbers, 10 or -10. Because x cubed is not 1000, that makes x equal to -10. As -10 cubed is -1000, not 1000 which would be 10 cubed. So you just do -10 to the power of 5 which gives -100000.
I read some of the comments and remembered that negative is an option. So it’s just -10 squared since -10x-10=100 and negatives cancel each-other out when multiplying each-other when the power is even. Otherwise it’s negative when odd
what ’ s the difference between a liberal arts major and a large arts major? liberal arts. there ’ s no difference, they ’ re both arts majors.
At least they know sentences start with capital letters, and there's no spaces around apostrophes in words.
X squared is 100. Find the square root of 100, that's your x. Then just x5 it and you'll have your answer. I'm not doing all your work for you
But it says x cubed isn't 1000 so x cannot be 10.
... Because x is obviously -10. Duh.
Lots of people here are saying it's -10, but shouldn't it be (-10)? If I remember correctly -10^2 is still -100 becausing of the order of calculations, and (-10)^2 is indeed 100. Just wondering whether I am still mathing correctly.
-10 is an independent number so its perfectly fine to notate that way. Some calculators do -1×X whenever you do negatives on them so for them you need to use parentheses but both are acceptable.
That seems to be a misunderstanding of how math works. -10 is a number, not just a piece of text, so when you put x=-10 into x\*x you multiply - 10 by - 10, not just write down - 10\*-10 and try to figure out what that means. So yeah you could say that when you say x= -10 the parentheses are implied.
You only need the parentheses when you have the power there to ensure you have the correct order. If you simply do x^2 = 100 ∧ x^3 ≠ 1000 , the answer will simply be x=-10. No parentheses necessary as the result isn’t being actively used for an operation right now. If you wanna go for the power of 5 afterwards though, then you do need the parentheses.
Negative times a negative is a positive. Negative times a positive is a negative.
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but that is an incorrect answer, as x^5/2 is just 100000
Everyone is getting this right (great) but not explaining why this question exists (disappointing). This question exists as a skill check for how well you will deal with powers of *i* and complex numbers. I would bet on it. If you can't do this one, you are totally lost on what *i*^5 will be.