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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?

Michael Hardy
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Ray Kay
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4 Answers4

18

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

kobe
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    Exponents have a higher precedence than negation. Think of it as $0-5^2$ (exponents come before subtraction). – Cole Tobin Mar 09 '15 at 21:24
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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.

graydad
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  • Looked up the pneumonic. It means "parentheses, exponents, multiplication, division, addition, subtraction". That is awful and I want to kill whoever made it. – bjb568 Mar 09 '15 at 21:08
  • bjb I agree. That pneumonic does more bad than good – imranfat Mar 09 '15 at 21:10
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    This is somewhat tangential, but the word is 'mnemonic', not 'pneumonic'. – Unochiii Mar 09 '15 at 21:11
  • @Unochiii I'll just remove everything in my post I said about the mnemonic and stick to what I know is right :) – graydad Mar 09 '15 at 21:12
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    @graydad, you should have left it -- you breathed new life into an old word! – Barry Cipra Mar 09 '15 at 21:15
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    I mean if you view that particular mnemonic as a harmful plague-like teaching that many young folk have drilled into their heads, pneumonic is pretty fitting! – graydad Mar 09 '15 at 21:18
6

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 $(-1)\cdot 25 = -25$.

Cameron Williams
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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.

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

John-Luke
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