NSW HSC Mathematics Advanced

Index laws

For a base $a \neq 0$ and whole-number or fractional indices, the laws follow from what an index counts: $a^m$ is $m$ copies of $a$ multiplied together, so combining them just counts copies.

Dividing $a^n$ by itself gives $1$ by the second law and $a^{n-n} = a^0$, so $a^0 = 1$ for $a \neq 0$. Dividing $a^0$ by $a^n$ in the same way gives a negative index:

A fractional index is a root. Since $\left(a^{1/n}\right)^n = a$, the number $a^{1/n}$ must be the $n$th root of $a$, and combining this with the power law gives

Taking the root first keeps the numbers small, so $8^{2/3} = \left(\sqrt[3]{8}\right)^2 = 2^2 = 4$, and $16^{-3/4} = \dfrac{1}{\left(\sqrt[4]{16}\right)^3} = \dfrac{1}{8}$.

Worked example: simplify $\dfrac{(2x^3)^2 \times x^{-5}}{\sqrt{x}}$.

An equation with the unknown in the index is solved by writing both sides with the same base. To solve $4^{x} = 32$, write $4 = 2^2$ and $32 = 2^5$, so $2^{2x} = 2^5$, which gives $2x = 5$ and $x = \tfrac{5}{2}$.

Expanding, factorising and simplifying

Expanding multiplies every term in one bracket by every term in the other. The special products come up so often that they are worth recognising on sight:

Factorising reverses expanding, and there is a fixed order to try. Take out any common factor first; then look for a difference of two squares; then factorise a quadratic trinomial; and for four terms try grouping in pairs.

For a trinomial $ax^2 + bx + c$ with $a \ne 1$, find two numbers whose product is $ac$ and whose sum is $b$, split the middle term with them, and group. For $6x^2 + 7x - 3$ the product $ac$ is $-18$ and the pair that sums to $7$ is $9$ and $-2$, so

Sums and differences of cubes also factorise, and are worth knowing for simplifying fractions:

Algebraic fractions

An algebraic fraction is handled exactly like a numerical one, with factorising doing the work that finding common factors does for numbers. Two points about validity matter. A denominator can never be zero, so any value of the variable that makes it zero is excluded, and cancelling a common factor is legitimate only when that factor is non-zero.

Simplifying, multiplying and dividing

Factorise every numerator and denominator, then cancel common factors. To divide, multiply by the reciprocal of the second fraction.

Here $x \neq -1, \, -3, \, 1$, because each of these makes a denominator, or the divisor, zero at some stage.

Adding and subtracting

Factorise the denominators, build the lowest common denominator from every distinct factor, rewrite each fraction over it, and combine the numerators.

Expressions such as $\dfrac{a^3b - ab^3}{a^2 + 2ab + b^2}$ simplify by taking out $ab$ in the numerator and recognising a square in the denominator:

Surds

A surd is an irrational root that cannot be written without a root sign, such as $\sqrt{2}$ or $\sqrt[3]{5}$; $\sqrt{9}$ is not a surd because it equals $3$. The rules for square roots of non-negative numbers are

and there is no rule for $\sqrt{a+b}$: it is not $\sqrt{a}+\sqrt{b}$. To simplify a surd, pull out the largest square factor, so $\sqrt{72} = \sqrt{36 \times 2} = 6\sqrt{2}$. Like surds add and subtract as like terms do: $3\sqrt{2} + 5\sqrt{2} = 8\sqrt{2}$.

Brackets containing surds expand as usual, and a surd multiplied by itself becomes rational:

Conjugates and rationalising the denominator

The conjugate of $\sqrt{a} + \sqrt{b}$ is $\sqrt{a} - \sqrt{b}$, and the other way round. Their product is a difference of two squares, so the roots disappear:

To rationalise a denominator, multiply the numerator and denominator by the conjugate of the denominator. When the denominator is a single surd, multiply by that surd instead.

When the numerator also contains surds, the same multiplication applies and the numerator is expanded:

The same method works when a fraction has two terms in its denominator and the numbers are not whole: the conjugate is formed from the same two terms with the sign between them changed.