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. These skills give exact answers, which Methods prefers over decimal approximations, and they recur whenever a calculation produces an irrational result or a fractional power.

Surd basics

A surd is an irrational root, such as $\sqrt 2$, that cannot be reduced to a rational number. Surds obey the multiplication and division rules

valid for $a, b \geq 0$. To simplify a surd, extract the largest perfect-square factor: $\sqrt{72} = \sqrt{36 \cdot 2} = 6\sqrt 2$. There is no corresponding rule for sums, so $\sqrt{a + b}$ does not split into $\sqrt a + \sqrt b$, a distinction worth stating explicitly because it is a frequent error.

Adding and subtracting surds

Surds add and subtract only when they are "like", that is share the same radicand after simplifying. So $3\sqrt 2 + 5\sqrt 2 = 8\sqrt 2$, but $\sqrt 2 + \sqrt 3$ cannot be combined. Often two surds look unlike but become like after simplification, for example $\sqrt{8} + \sqrt{18} = 2\sqrt 2 + 3\sqrt 2 = 5\sqrt 2$, so always simplify each surd before deciding whether terms 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

The index laws combine powers that share a base, and they are the backbone of all exponent simplification:

The first move in almost any simplification is to express every quantity as a power of a common base (for example writing $8$ as $2^3$), because the laws only act on matching bases. Surds fit in through $\sqrt a = a^{1/2}$, so a surd expression can always be recast as an index expression and simplified with the same laws.

Rational and negative exponents

Fractional and negative powers extend the index laws so that roots and reciprocals fit the same framework:

The denominator of a fractional index is the root and the numerator is the power, so $8^{2/3} = (\sqrt[3]{8})^2 = 2^2 = 4$ and $16^{3/4} = (\sqrt[4]{16})^3 = 2^3 = 8$, while $5^{-2} = \tfrac{1}{25}$. This unification is what lets a single set of laws simplify any expression mixing roots, reciprocals and powers, which is the recurring task in this dot point.

Why rationalise with the conjugate

For a binomial surd denominator such as $\sqrt 5 - \sqrt 2$, multiplying by the conjugate $\sqrt 5 + \sqrt 2$ exploits the difference of squares: $(\sqrt 5 - \sqrt 2)(\sqrt 5 + \sqrt 2) = 5 - 2 = 3$, a rational number. The conjugate is the same two terms with the middle sign flipped, and this is the only multiplier that clears both surds at once. The same idea evaluates expressions of the form $(a + \sqrt b)(a - \sqrt b) = a^2 - b$.

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.