Simplify expressions involving surds and apply the laws of indices to rational and negative exponents

What this dot point is asking

QCAA wants you to simplify expressions involving surds (including rationalising denominators) and apply the laws of indices to rational and negative exponents.

Surd basics

A surd is an irrational root that cannot be simplified to a rational number.

$\sqrt a \cdot \sqrt b = \sqrt{ab}$, $\dfrac{\sqrt a}{\sqrt b} = \sqrt{a/b}$.

Simplify by removing perfect-square factors. $\sqrt{72} = \sqrt{36 \cdot 2} = 6\sqrt 2$.

Adding and subtracting surds

Like terms only. $3\sqrt 2 + 5\sqrt 2 = 8\sqrt 2$. Unlike surds do not combine.

Rationalising denominators

Eliminate surds from the denominator.

Monomial: $\dfrac{1}{\sqrt 3} = \dfrac{\sqrt 3}{3}$.

Binomial: multiply by conjugate. $\dfrac{1}{\sqrt 5 - 1} = \dfrac{\sqrt 5 + 1}{4}$.

Index laws

$a^m a^n = a^{m+n}$, $\dfrac{a^m}{a^n} = a^{m-n}$, $(a^m)^n = a^{mn}$, $(ab)^n = a^n b^n$, $\left(\dfrac{a}{b}\right)^n = \dfrac{a^n}{b^n}$, $a^0 = 1$ ($a \neq 0$), $a^{-n} = \dfrac{1}{a^n}$.

Rational and negative exponents

$a^{1/n} = \sqrt[n]{a}$. $a^{m/n} = (\sqrt[n]{a})^m = \sqrt[n]{a^m}$.

$5^{-2} = 1/25$. $8^{2/3} = 4$. $16^{3/4} = 8$.

Worked example

Simplify $\dfrac{(2x^2)^3 \cdot x^{-1}}{4 x^{3/2}}$.

Numerator: $8 x^6 \cdot x^{-1} = 8 x^5$.

Divide: $\dfrac{8 x^5}{4 x^{3/2}} = 2 x^{5 - 3/2} = 2 x^{7/2}$.

Common traps

Adding unlike surds. $\sqrt 2 + \sqrt 3 \neq \sqrt 5$.

Sign on $-a^2$. $-3^2 = -9$, but $(-3)^2 = 9$.

Forgetting to rationalise. QCAA expects rational denominators.

In one sentence

Surds simplify by extracting perfect-square factors and rationalising denominators (monomial: multiply by the surd; binomial: multiply by the conjugate); the index laws extend to rational ($a^{m/n} = \sqrt[n]{a^m}$) and negative ($a^{-n} = 1/a^n$) exponents.