Unable To Understand The Difference Between $(-3)^4$ And $-3^4$

Question

Why is $(-3)^4 =81$ and $-3^4 =-81 $?This might be the most stupidest question that you might have encountered,but unfortunately i'am unable to understand this.

Answer 1

By definition

$(-3)^4 = (-3)\cdot(-3)\cdot(-3)\cdot(-3) = 9\cdot 9 = 81$

$-3^4 = -(3^4) = -(3\cdot 3\cdot 3\cdot 3) = - 81$

Answer 2

Note that $(-3)^4=((-1)\cdot3)^4=(-1)^4\cdot3^4=81$

Again note that $-3^4=(-1)\cdot3^4=-81$

Answer 3

$$(-3)^4 = -3 \cdot -3\cdot -3\cdot -3 = 9\cdot9=81 $$Now, when we have to "simplify" the expression $-3^4$, first we do the exponent and then the rest. So this simplifies to $-(81) = -81$

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This question had confused me a lot too, and then I got my enlightenment (now waiting for the badge)...

Answer 4

Because multiplication doesn't distribute into exponentiation. That is, $a(b^c) = (ab)^c$ doesn't always hold. And there's no reason why it should, unless $c = 1$. So $(-3)^4 = (-1 \cdot3)^4 = (-1)^4 \cdot (3)^4 = 81$, and not $-81$, as the multiple of $-1$ doesn't factor out.