Integer Exponents

An exponent is simply shorthand for "multiply this number by itself so many times": $2^5$ is five 2's multiplied together. "Integer" just means the exponent is a whole number, which here includes zero and negatives. It's good for writing huge and tiny numbers compactly and for anything that grows by repeated doubling or tripling. The rules up top look like a wall of symbols, but each is a small, common-sense statement: $a^ma^n = a^{m+n}$ (put two piles of factors together and count them all), $\frac{a^m}{a^n} = a^{m-n}$ (dividing cancels some of them) and $(a^m)^n = a^{mn}$ (a pile of piles). Bear with it; it's a few simple ideas.

Why the rules hold. You don't need to memorize these; you can rebuild them any time by writing out the factors. Take $a^2a^3$. That's $(a\cdot a)(a\cdot a\cdot a)$, which is five $a$'s in a row, so $a^5$. The exponents add because you're just counting factors. Dividing cancels factors from the top against the bottom, so the exponents subtract.

Now think about $\frac{a^3}{a^3}$. Anything divided by itself is $1$. But the subtraction rule says it's also $a^{3-3} = a^0$. Both have to be true, so $a^0 = 1$. Nobody decided that by decree; it's the only value that keeps the rules consistent.

Negative exponents come from the same move. $\frac{a^2}{a^5}$ leaves three $a$'s on the bottom, so it's $\frac{1}{a^3}$, and the subtraction rule calls it $a^{-3}$. So a negative exponent means "on the other side of the fraction bar": $x^{-2} = \frac{1}{x^2}$ and $\frac{1}{y^{-3}} = y^3$. Notice it's a reciprocal, not a negative number.

A common mistake. It's tempting to write $(a + b)^2 = a^2 + b^2$, because $(ab)^2 = a^2b^2$ works. It doesn't: exponents spread over multiplication, not addition. Try $a = b = 1$: $(1 + 1)^2 = 4$, but $1^2 + 1^2 = 2$. Another one to watch is $-3^2$. The exponent grabs only the $3$, so $-3^2 = -9$, while $(-3)^2 = 9$.

Scientific notation ($6.02\times10^{23}$), data sizes ($2^{10}$ bytes in a kilobyte) and compound growth all run on exponent rules.