Why is $-5^2=-25$?

Question

If $-5^2$ is equal to $(-5)(-5)$, doesn't that mean the negatives should cancel each other out and become $25$? Why is this not the case?

Answer 1

You're confusing $(-5)^2$ with $-5^2$. We have $(-5)^2 = (-5)(-5) = 25$, but $-5^2 = -(5^2) = -25$.

Answer 2

Parentheses are your friend. $-5^2$ actually means $-(5^2) = -25$. $-5^2$ does not mean $(-5)^2$. Think order of operations: parentheses and exponents first, then multiplication and division, then addition and subtraction. You can view $-5^2$ as being $(-1)\cdot 5^2$. Doing exponents first, you get $(-1)\cdot 25 = -25$.

Answer 3

I'm guessing you calculated this on a calculator. Since exponentiation is ranked higher than multiplication as far as order of calculation is concerned, the calculator reads $-1\cdot 5^2$, and so calculates the $5^2$ portion before multiplying by $-1$. This is remedied by wrapping $-5$ in parentheses before evaluating. $(-5)^2$ should give you the answer you would expect.

Answer 4

Because you forgot the parentheses. You meant to calculate $(-5)^2$ but instead calculated $-(5^2)$. When in doubt, use more parentheses than seem necessary.

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By the way, this also applies to imaginary numbers: $(-5i)^2 = -25$, but $-(5i)^2 = 25$.