Why does $(-2^2)^3$ equal $-64$ and not $64$?

The title says it all. Why does $(-2^2)^3$ equal $-64$ and not $64$? This was on my algebra final, and I am completely stuck on how it works.

• I always have an issue with this because it is somewhat of an ambiguous notation. I think that when doing mathematics, that the concepts should be tested and not the differing interpretations of a problem. – yousuf soliman Jun 19 '13 at 19:22

The negative sign ($-$) applies to the quantity $2^2$, so that $-2^2$ means $-(2^2)=-4$, not $(-2)^2=4$. $$(-2^2)^3=(-4)^3=-64$$

• So for it to actually be 64, would the problem need parentheses around the -2? So would it have to be $((-2)^2)^3$ for it to equal 64? – S17514 Jun 19 '13 at 19:19
• Yup, that's right. – Zev Chonoles Jun 19 '13 at 19:20

Note that $-2^2 = - (2 \times 2) = -4$ and is not $(-2) \times (-2) = 4$. Hence, $$(-2^2)^3 = (-4)^3 = (-4) \times (-4) \times (-4) = - 64$$

In general, when $m$ is a positive integer, we have $$-a^m = - (\underbrace{a \times a \times \cdots \times a}_{m \text{times}})$$ and is not $$(-a)^m = (\underbrace{(-a) \times (-a) \times \cdots \times (-a)}_{m \text{times}})$$

• It's worth noting that some simplistic tokenizers do parse $-2^2$ as $(-2)^2$, which is non-standard. I believe most spreadsheets are guilty of this (perhaps to stay compatible with Microsoft Excel). – Erick Wong Jun 19 '13 at 19:21

Rewrite it as

$(-4)^3=(-1)^3 \cdot 4^3=-64$