Powers & Exponents

A power like $3^4$ has a base ($3$) and an exponent ($4$). The exponent counts how many times the base is multiplied by itself:

$3^4 = 3 \times 3 \times 3 \times 3 = 81$

The most common mistake is multiplying the base by the exponent — $3^4$ is not $3 \times 4 = 12$. It's four factors of $3$.

Two squares to keep straight with negative bases: a negative to an even power is positive (the negatives pair up and cancel), and to an odd power stays negative. Parentheses make this visible — $(-2)^4$ means $(-2)$ used four times.

Build up one factor at a time. For $2^5$, double five times:

$2^5 = 2 \times 2 \times 2 \times 2 \times 2 = 32$

Powers of ten are just a $1$ followed by that many zeros — $10^3 = 1000$.

Negative bases — count the signs. An even exponent pairs the negatives off:

$(-3)^2 = (-3)(-3) = 9$

An odd exponent leaves one negative behind:

$(-2)^3 = (-2)(-2)(-2) = -8$

A negative base is where signs go wrong. Count how many negatives you have, then decide the sign. Predict before you check.