6.1 Laws of indices
Everything in this chapter rests on a handful of rules for powers. For \(a>0\) and any real \(m,n\):
| Rule | Statement | Example |
| Product | \(a^m\times a^n=a^{m+n}\) | \(5\times 5^2=5^3=125\) |
| Quotient | \(a^m\div a^n=a^{m-n}\) | \(y^9\div y^5=y^4\) |
| Power of a power | \(\left (a^m\right )^n=a^{mn}\) | \(\left (2^3\right )^2=2^6=64\) |
| Negative index | \(a^{-m}=\frac {1}{a^m}\) | \(2^{-2}=\frac {1}{4}\) |
| Fractional index | \(a^{m/n}=\sqrt [n]{a^m}\) | \(8^{2/3}=\left (\sqrt [3]{8}\right )^2=4\) |
| Zero index | \(a^0=1\) | \(7^0=1\) |
Note 6.1. The rules for products and quotients apply only when the bases match. There is no rule for \(2^3\times 3^2\); those must simply be worked out separately. Trying to add the indices of different bases is a frequent and expensive error.
Questions on this section
Stuck on something here? Ask below and it stays attached to this topic.