TMUA (Test of Mathematics for University Admission)

## Laws of indices An index (or exponent) says how many copies of a base are multiplied. For a positive base $a$ and any rational exponents $m$ and $n$ the laws are $$ a^m \times a^n = a^{m+n} \qquad \frac{a^m}{a^n} = a^{m-n} \qquad (a^m)^n = a^{mn} $$ and two further results extend the idea of a power beyond positive whole numbers: $$ a^0 = 1 \qquad a^{-n} = \frac{1}{a^n} $$ The zero power is $1$ because $a^n \div a^n = a^{n-n} = a^0$ and anything divided by itself is $1$. A negative power is a reciprocal, so $2^{-3} = \tfrac{1}{8}$ and $\left(\tfrac{2}{3}\right)^{-2} = \tfrac{9}{4}$: the fraction is turned upside down and the exponent made positive. ### Fractional exponents A power of $\tfrac{1}{n}$ is an $n$th root, and a general fractional power combines a root with a power: $$ a^{1/n} = \sqrt[n]{a} \qquad a^{m/n} = \left(\sqrt[n]{a}\right)^m = \sqrt[n]{a^m} $$ Take the root first when the numbers are whole, because it keeps them small. $$ \begin{aligned} 27^{2/3} &= \left(\sqrt[3]{27}\right)^2 = 3^2 = 9 \\ 16^{-3/4} &= \frac{1}{\left(\sqrt[4]{16}\right)^3} = \frac{1}{2^3} = \frac{1}{8} \end{aligned} $$ ### Worked example Write $\dfrac{8^{2/3} \times 2^{-1}}{4^{1/2}}$ as a single power of $2$. Express every base as a power of $2$: $8 = 2^3$ and $4 = 2^2$. Then $$ \begin{aligned} \frac{(2^3)^{2/3} \times 2^{-1}}{(2^2)^{1/2}} &= \frac{2^{2} \times 2^{-1}}{2^{1}} \\ &= \frac{2^{1}}{2^{1}} \\ &= 2^0 = 1 \end{aligned} $$ Converting to a common base turns products and…