document.write( "Question 233405: Regarding negative exponents. Please explain why 2^-2 evaluates to 1/2^2 =1/4, but (-2^3)^2 evaluates to 2^6=64. I do not understand why the second expression results in a whole number. bljones1944@charter.net \n" ); document.write( "

Algebra.Com's Answer #172272 by jsmallt9(3758) $\"\"$ $\"About$

You can put this solution on YOUR website!
The key difference between your two examples is that in $\"2%5E%28-2%29\"$ the minus sign is on the exponent and in $\"%28-2%5E3%29%5E2\"$ the minus sign is not in an exponent. A minus sign in an exponent means \"reciprocal of\". It does not mean that a negative number is being multiplied anywhere.
\n" ); document.write( "P.S. $\"%28-2%5E3%29%5E2+=+%28-%282%5E3%29%29%5E2+=+%28-%282%2A2%2A2%29%29%5E2+=+%28-8%29%5E2+=+64\"$ . The only way to get $\"2%5E6\"$ while simplifying is an unusual path:
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\n" ); document.write( "The important point here is that $\"-2%5E3\"$ means $\"-%282%2A2%2A2%29\"$ . The exponent applies only to the 2, not to the minus sign. On the other hand, $\"%28-2%29%5E3\"$ means $\"%28-2%29%28-2%29%28-2%29\"$ . The difference may seem trivial because with an odd exponent like 3 the two work out the same. But with even exponents, the difference is critical.

\n" ); document.write( "Exponents apply only to what is immediately in front of them. In the first case, the first thing in front of the 3 is 2. In the second case, it is \")\" in front of the 3 (which means the exponent applies to the entire expression in the parentheses). \n" ); document.write( "

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