Simplify and operate on surd expressions and apply the laws of indices to rational and negative exponents

What this dot point is asking

VCAA wants you to simplify and operate on surds (including rationalising denominators) and to apply the index laws to rational and negative exponents.

Surd basics

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

Surds are simplified 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: $\sqrt 2 + \sqrt 3$ stays as is.

Rationalising denominators

Eliminate surds from a denominator.

Monomial denominator. Multiply top and bottom by the surd: $\dfrac{1}{\sqrt 3} = \dfrac{\sqrt 3}{3}$.

Binomial denominator. Multiply by the conjugate: $\dfrac{1}{\sqrt 5 - 1} = \dfrac{\sqrt 5 + 1}{(\sqrt 5 - 1)(\sqrt 5 + 1)} = \dfrac{\sqrt 5 + 1}{4}$.

Rational exponents

Example: $8^{2/3} = (\sqrt[3]{8})^2 = 2^2 = 4$.

Negative exponents

Example: $5^{-2} = 1/25$.

All index laws apply

The seven index laws ($a^m a^n = a^{m+n}$, etc.) apply to rational and negative exponents:

$x^{1/2} \cdot x^{1/3} = x^{1/2 + 1/3} = x^{5/6}$.

$(x^{2/3})^3 = x^2$.

Worked example

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

Numerator: $64 x^{3/2} \cdot x^{-1} = 64 x^{1/2}$.

Divide: $\dfrac{64 x^{1/2}}{2 x^{3/2}} = 32 x^{1/2 - 3/2} = 32 x^{-1} = \dfrac{32}{x}$.

Common traps

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

Forgetting to rationalise. Most exam answers expect the denominator to be rational.

Confusing $a^{-n}$ with $-a^n$. $5^{-2} = 1/25$, not $-25$.

Mixing rational exponent rules. $a^{m/n}$ is the $n$th root raised to the $m$th power, not the $m$th root raised to the $n$th power... actually these are equal because $(a^m)^{1/n} = a^{m/n} = (a^{1/n})^m$. But be careful with negative $a$: $(-8)^{1/3} = -2$ but $(-8)^{2/6}$ is ambiguous.

In one sentence

Surds are simplified by extracting perfect-square factors ($\sqrt{72} = 6\sqrt 2$), like surds add and subtract (unlike surds do not), denominators are rationalised by multiplying by the surd or its conjugate, and the index laws extend to rational ($a^{m/n} = \sqrt[n]{a^m}$) and negative ($a^{-n} = 1/a^n$) exponents.