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Calculating $81^{3/2}$, I got $729$ (not saying it is correct, but I am trying :) ). Would $-81^{3/2}$ just be the opposite ($-729$) and does it make a difference if $-81$ was placed inside a pair of parentheses $(-81)^{3/2}$?

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    $\begingroup$ This tutorial explains how to typeset mathematics on this site. $\endgroup$ Sep 15 at 0:26
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    $\begingroup$ See Powers and Roots of Complex numbers. Also note that: $-2^3\ne (-2)^3=(-1)^3 2^3$ $\endgroup$ Sep 15 at 0:28
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    $\begingroup$ @TymaGaidash When I put both into the calculator I get the same product (-8). Am I doing something wrong? $\endgroup$
    – יהודה
    Sep 15 at 0:30
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    $\begingroup$ @יהודה You are right, Wait that was a bad example, it does not matter for odd integers, try doing: $-4=-2^2\ne (-2)^2=(-1)^2 2^2=4$ instead. $\endgroup$ Sep 15 at 1:22
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    $\begingroup$ $(-81)^\frac{3}{2}=(9i)^3=-729i$ $\endgroup$ Sep 15 at 1:46
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We can think of the $-$ sign as the following words: "the opposite of".

In this case, $-1$ is the opposite of $1$, which just means it's the number that, when added to $1$, results in $0$.

Similarly, $-\pi$ is the opposite of $\pi$, the number that, when added to $\pi$, results in $0$.

In your question, $-81^{3/2}$ is a negative number, the number that, when added to $81^{3/2}$, results in $0$. You can simplify $81^{3/2}=729$ to see that $-81^{3/2}$ then must be $-729$.

However, $(-81)^{3/2}$ has the $-$ sign inside the parentheses. This means that the "opposite of" quality applies to $81$ and THEN we apply an exponent. Using properties of exponents, we do get that this is equal to $(-1)^{3/2}(81)^{3/2}$. If you are unfamiliar with complex numbers, then just saying that you can't take the square root of the $-1$ is sufficient for this, and is enough to see that this is very different than the original $-81^{3/2}$.

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I'm also new to Mathematics but I will try and explain in simple terms:

Using exponent rules we can say that $\left(-81\right)^{\frac{3}{2}}$ is equal to $\sqrt[2]{\left(-81\right)^{3}}$. In this case $\left(-81\right)^{3}$ will give us a negative number, so calculating $\left(-81\right)^{\frac{3}{2}}$ is only possible with the use of imaginary numbers.

It does make a difference if you put it in parentheses. If you type $-81^{\frac{3}{2}}$ in a calculator it will give you the anwer $-729$; this is only because it determines the expression as $-\left(81^{\frac{3}{2}}\right)$ meaning it will first evaluate the expression $\sqrt[2]{81^{3}}$ which will result in a positive number and then multiply it by $-1$.

More info: https://www.khanacademy.org/math/cc-eighth-grade-math/cc-8th-numbers-operations/exponents-with-negative-bases/v/exponents-with-negative-bases

Cheers, Tom

(I am a math 'noob' so if I've missed something please let me know.)

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  • $\begingroup$ Thank you, Tom. So within the parenthesis negative number (-a) with an exponent (-a), ^x will be classified as imaginary. Copy that. $\endgroup$
    – יהודה
    Sep 15 at 2:24
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    $\begingroup$ @יהודה That's exactly right, hope you are satisfied with the answer :) $\endgroup$
    – Tom Joney
    Sep 15 at 2:33
  • $\begingroup$ I am and thank you again. Even though you claim that you are new at this, it seems as though you have a great grasp of the concept. Keep it up!!! $\endgroup$
    – יהודה
    Sep 15 at 2:40

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