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

and two further results extend the idea of a power beyond positive whole numbers:

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:

Take the root first when the numbers are whole, because it keeps them small.

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

Converting to a common base turns products and quotients into sums and differences of exponents, which is the quickest way to handle mixed powers. The same idea solves equations such as $4^x = 8$: writing $2^{2x} = 2^3$ gives $x = \tfrac{3}{2}$.

Conditions

The laws need a positive base. A negative base with a fractional exponent can be undefined in the real numbers: $(-8)^{1/3} = -2$ is a real number, but $(-4)^{1/2}$ is not, and $\left((-1)^2\right)^{1/2} = 1$ is not the same as $(-1)^{2 \times 1/2} = -1$. So $(a^m)^n = a^{mn}$ is used freely only when $a > 0$.

Surds

A surd is an irrational root, such as $\sqrt{2}$ or $\sqrt{5}$. Its decimal never ends and never repeats, so an exact answer keeps the root symbol. The two rules for square roots of positive numbers are

Note that $\sqrt{a+b}$ is not $\sqrt{a}+\sqrt{b}$: for instance $\sqrt{9+16} = 5$ but $\sqrt{9}+\sqrt{16} = 7$.

Simplifying

Pull out the largest square factor. Then $\sqrt{75} = \sqrt{25 \times 3} = 5\sqrt{3}$ and $\sqrt{450} = \sqrt{225 \times 2} = 15\sqrt{2}$. Surds can be added or subtracted only when they are the same root after simplifying:

Expanding brackets

Expand as normal and use $\sqrt{a}\times\sqrt{a} = a$.

The second is a difference of two squares, and the answer is rational because the cross terms cancel.

Rationalising the denominator

A fraction with a surd in the denominator is rewritten so that the denominator is a whole number.

For a single surd, multiply top and bottom by that surd:

For a denominator of the form $a + b\sqrt{c}$, multiply top and bottom by its conjugate $a - b\sqrt{c}$, so that the denominator becomes the difference of two squares $a^2 - b^2c$.

A second example, where both parts of the denominator contain surds:

Surds inside larger problems

Rationalising is also the route to simplifying a sum of fractions. For example

and to showing that an expression is rational: $(1+\sqrt{2})^3 + (1-\sqrt{2})^3$ is unchanged by swapping the sign of $\sqrt{2}$, so every odd power of $\sqrt{2}$ cancels and only a rational number remains. Expanding gives $2(1 + 3\cdot 2) = 14$.