# What's the difference between $-81^{3/2}$ & $(-81)^{3/2}$?

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 $$(-81)^{3/2}$$?

• This tutorial explains how to typeset mathematics on this site. Commented Sep 15, 2021 at 0:26
• See Powers and Roots of Complex numbers. Also note that: $-2^3\ne (-2)^3=(-1)^3 2^3$ Commented Sep 15, 2021 at 0:28
• @TymaGaidash When I put both into the calculator I get the same product (-8). Am I doing something wrong? Commented Sep 15, 2021 at 0:30
• @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. Commented Sep 15, 2021 at 1:22
• $(-81)^\frac{3}{2}=(9i)^3=-729i$ Commented Sep 15, 2021 at 1:46

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 $$-81^{3/2}$$.

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 $$-1$$.

Cheers, Tom

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

• Thank you, Tom. So within the parenthesis negative number (-a) with an exponent (-a), ^x will be classified as imaginary. Copy that. Commented Sep 15, 2021 at 2:24
• @יהודה That's exactly right, hope you are satisfied with the answer :) Commented Sep 15, 2021 at 2:33
• 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!!! Commented Sep 15, 2021 at 2:40