Question

Explain why –2^4 and (–2)^4 are different when they are evaluated, but –2^3 and (–2)^3 are not.

Answer 1

The answer here has to do with order of operations rules (PEMDAS).

–2^4 comes out to -16, because the exponentiation MUST be done first. That gives you 16, which you then preface with the - symbol.

(–2)^4 Here the parentheses make the difference: (-2)^4 requires that we take the 4th power of (-1) and mult. that by the 4th power of 2, which is 16.

So, yes, –2^4 and (–2)^4 are different.

What about –2^3 and (–2)^3?

Again we must do exponentiation first. 2^3 is 8, which we then preface with -. Result: -8. (–2)^3 is equivalent to (1)^3 * 2^3, or (-1)*8, or -8. So these two results are identical.

Perhaps it would help to recall that

-1 = -1

(-1)^2 = 1

(-1)^3 = -1

(-4)^4 = 1

and that this pattern repeats itself after you've written each group of 4 lines.