The Laws of Exponents

A power is repeated multiplication: $a^n$ means $a$ multiplied by itself $n$ times. The $a$ is the base, the $n$ is the exponent (or index). Three laws let you combine powers of the same base without writing everything out.

Product law — multiplying, so add the exponents:

Quotient law — dividing, so subtract the exponents:

Power of a power — multiply the exponents:

Worked example. $x^3 \times x^4 = x^{3+4} = x^7$, and $(2^3)^2 = 2^{6} = 64$. Why does adding work? Because $x^3 \times x^4$ is just $(x\cdot x\cdot x)\times(x\cdot x\cdot x\cdot x)$ — seven $x$'s in a row.