Index laws with negative and fractional indices

Indices — the basics

• An index (or power / exponent) shows repeated multiplication: $a^n$ means $a$ multiplied by itself $n$ times. The base is $a$.
• The index laws let you simplify powers of the same base without writing them out.

The three main laws

• Multiplying — add the indices: $a^m \times a^n = a^{m+n}$. (You are just counting the total number of factors: $a^2 \times a^3 = (aa)(aaa) = a^5$.)
• Dividing — subtract the indices: $a^m \div a^n = a^{m-n}$. (Common factors cancel, so the count drops.)
• Power of a power — multiply the indices: $(a^m)^n = a^{mn}$.
• Also useful: $(ab)^n = a^n b^n$ — a power outside a bracket applies to everything inside.

Zero and negative indices

• Zero index: $a^0 = 1$ for any non-zero $a$. (This follows the dividing law: $\dfrac{a^3}{a^3} = a^{3-3} = a^0$, but anything over itself is 1.)
• Negative index means "one over" — the reciprocal: $a^{-n} = \dfrac{1}{a^n}$.
  • So $2^{-3} = \dfrac{1}{2^3} = \dfrac{1}{8}$.
  • This too comes from the dividing law: $\dfrac{a^2}{a^5} = a^{2-5} = a^{-3} = \dfrac{1}{a^3}$.

Fractional indices are roots

• The denominator of a fractional index is a root: $a^{1/n} = \sqrt[n]{a}$.
  • Why? $a^{1/2} \times a^{1/2} = a^{1} = a$, and the number that times itself gives $a$ is $\sqrt{a}$ — so $a^{1/2} = \sqrt{a}$.
• In general the bottom is the root and the top is the power: $a^{m/n} = \left(\sqrt[n]{a}\right)^m = \sqrt[n]{a^m}$.
• So $8^{2/3} = \left(\sqrt[3]{8}\right)^2 = 2^2 = 4$.