Powers, roots and exponents

Every time your phone storage doubles , 64 GB, 128 GB, 256 GB , that's exponents doing the work. Instead of writing 2 × 2 × 2 × 2 × 2 × 2, you write 2⁶. That's the whole idea.

In the expression $a^n$, the base $a$ is the number being repeatedly multiplied, and the exponent $n$ (also called the index or power) tells you how many times to multiply it by itself:

$a^n=\underset{n\text{ factors}}{\underbrace{a\times a\times \cdots \times a}}$

So $5^3=5\times 5\times 5=125$, and $(-2{)}^5=(-2)\times (-2)\times (-2)\times (-2)\times (-2)=-32$.

Here's the thing that trips almost everyone up: where the negative sign sits completely changes the answer. Compare $-2^4$ with $(-2{)}^4$. In $-2^4$, the exponent only applies to the 2, so you get $-(2^4)=-16$. But in $(-2{)}^4$, the negative is part of the base, giving you $(-2)\times (-2)\times (-2)\times (-2)=+16$. Two expressions that look nearly identical, two completely different answers.