Laws of exponents

What the symbols mean

$P$

the product $a^m·a^n$

$a$

the base (positive in the calculator, so any real exponent works)

$m$

the first exponent

$n$

the second exponent

When to use it

Use the index laws to simplify expressions with powers, to rewrite roots and reciprocals as powers before differentiating or integrating, and to solve equations by matching bases: $2^{x+1}=2^5$ gives $x=4$.

Why it works

$a^m$ is m copies of a and $a^n$ is n copies; multiplied together that is m + n copies. Defining $a^0=1$, $a^{-n}=1/a^n$ and $a^{1/n}=\sqrt[n]{a}$ is exactly what keeps this law true for all exponents.

Common mistakes

• Multiplying the exponents when multiplying powers: $2^3·2^4=2^7$, not $2^{12}$.
• Adding exponents of different bases: $2^3·3^4$ does not simplify to $6^7$.
• Thinking $a^{-n}$ is negative. $2^{-3}=1/8$, which is positive.