Exponent Rules

Every exponent rule comes straight from "an exponent counts factors." Multiply $2^3 \times 2^2$ and you're just stacking factors of $2$:

$2^3 \times 2^2 = (2 \cdot 2 \cdot 2)(2 \cdot 2) = 2^5$

Three factors plus two more makes five — so for the same base, multiplying means adding exponents. Dividing cancels factors, so it means subtracting. And a power of a power, like $(2^3)^2$, is two copies of $2^3$ multiplied — $2^3 \times 2^3 = 2^6$ — so it means multiplying the exponents.

The catch that trips everyone: these rules combine the exponents, never the bases. $3^2 \times 3^4 = 3^6$, not $9^6$ — the base stays $3$.

Product — add the exponents.

$3^4 \times 3^2 = 3^{4+2} = 3^6$

Quotient — subtract the exponents.

$5^7 \div 5^3 = 5^{7-3} = 5^4$

Power of a power — multiply the exponents.

$(2^3)^2 = 2^{3 \times 2} = 2^6$

Same base every time; only the exponents combine, by the operation that matches.

Decide which rule applies, then combine the exponents — and keep the base. Predict before you check.