What is the difference between squaring a negative number inside and outside of parentheses?

Question

What is the difference between: $ -3^2 $ and $ (-3)^2 $ ? I know $-3^2 = -9$ and $(-3)^2 = 9$ but I don't know why.

Because when you $-3^2$ you're not squaring -ve sign here. But when you use parentheses you square -ve sign as well. — Iti Shree, Apr 29 '17 at 18:35
$-3^2 = (-1) \cdot 3^2 = (-1) \cdot 9 = -9,$ because exponentiation has higher precedence than multiplication. — dxiv, Apr 29 '17 at 18:37
Related: What is the accepted syntax for a negative number with an exponent? — Andrew D. Hwang, Apr 29 '17 at 18:37
$(-3)^2=((-1)(3))^2=(-1)^2\cdot (3)^2=9$ while $-3^2=-(3)^2=-9$ — kingW3, Apr 29 '17 at 18:38

score 2 · Answer 1 · answered Apr 29 '17 at 18:36

It's a matter of the order of operations.

If you write $-3^{2}$, you're saying "the square of 3, now make it negative" = "9, now make it negative" = "-9".

If you write $(-3)^{2}$, you're saying "make 3 negative, now square it" = "-3, now square it" = "(-3)(-3)" = "9".

score 0 · Answer 2 · answered Apr 29 '17 at 18:36

So by convention, the power operation has higher priority than multiply operation unless you have "$()$" to indicate priority.

$$-3^2=-1\times 3^2=-1\times 3\times 3$$

While

$$(-3)^2=(-3)\times (-3)=(-1\times 3)\times (-1\times 3) = -1\times -1 \times 3\times 3$$

2 Answers2