Work with the real number system and surds: simplify surds, add, multiply and expand surdic expressions, rationalise single- and binomial-surd denominators, and apply the index laws to expressions with integer indices

What this dot point is asking

Before you can graph, differentiate or solve a quadratic with confidence, you have to be at home with the numbers themselves. NESA expects you to know the real number system, to recognise when a root is a surd (an irrational number that cannot be written as a fraction), and to manipulate surds fluently: simplify them, add and multiply them, expand brackets that contain them, and clear them out of a denominator. You also need the index laws cold, because powers are the language the rest of the course is written in. The methods are short, but they recur in almost every later topic, so the goal is speed without slips, and understanding why each rule holds so you can rebuild it under pressure.

The answer

The real number system: N, Z, Q and R

Mathematics builds its numbers up in layers, each one larger than the last:

• Natural (counting) numbers $\mathbb{N}$: $1, 2, 3, 4, \dots$ (and often $0$). The numbers you count with.
• Integers $\mathbb{Z}$: the naturals together with zero and the negatives, $\dots, -2, -1, 0, 1, 2, \dots$
• Rational numbers $\mathbb{Q}$: every number that can be written as a fraction $\dfrac{a}{b}$ of two integers (with $b \neq 0$). Equivalently, every terminating or recurring decimal. For example $\dfrac{3}{4} = 0.75$ and $\dfrac{3}{11} = 0.\overline{27}$ are rational.
• Real numbers $\mathbb{R}$: all the points on the number line. The reals that are not rational are the irrational numbers, such as $\sqrt{2}$, $\sqrt{3}$ and $\pi$. An irrational number has a decimal that never terminates and never recurs.

Each set sits inside the next: $\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R}$. The picture below places a sample number in the smallest set that contains it, so you can see at a glance where any given number lives, and that the irrationals are exactly the reals left over once you remove the fractions.

A subtle point worth keeping: the rationals are dense, meaning between any two of them there is always another, yet they still leave gaps. The number $\sqrt{2}$ sits in one of those gaps. It is the length of the diagonal of a unit square, so it certainly names a point on the line, but no fraction equals it exactly. That is what "irrational" means, and it is why we keep such numbers in exact form ($\sqrt{2}$, not $1.41$) until a decimal is actually required.

What a surd is, and why $\sqrt{2}$ is irrational

A surd is a root such as $\sqrt{2}$, $\sqrt[3]{5}$ or $\sqrt{7}$ that is not itself a rational number. The test is whether the root comes out exactly: $\sqrt{9} = 3$ and $\sqrt[3]{8} = 2$ are not surds because they simplify to integers, whereas $\sqrt{2}$ is a surd because it does not. Notice that $\sqrt{x}$ is the positive square root only: although $25$ has two square roots, $5$ and $-5$, the symbol $\sqrt{25}$ denotes just $5$, and we write $-\sqrt{25}$ for the negative one.

That surds really are irrational is not obvious, and a famous proof by contradiction shows it. The argument is worth seeing once because it explains why surds cannot be tidied into fractions; the worked example "Prove that $\sqrt{3}$ is irrational" below runs it in full.

Simplifying a surd: take out the largest square factor

A surd is in simplest form when the number under the root sign has no square factor bigger than $1$. To simplify, pull out the largest perfect square ($4, 9, 16, 25, 36, \dots$) that divides the number, using the law $\sqrt{ab} = \sqrt{a}\,\sqrt{b}$ for non-negative $a$ and $b$:

Always take out the largest square you can spot, or you will have to simplify twice. If you only notice $\sqrt{72} = \sqrt{4 \times 18} = 2\sqrt{18}$, you are not finished, because $18 = 9 \times 2$ still has the square factor $9$. The two laws that do all the work are

Surd arithmetic: collecting, multiplying and expanding

Once surds are simplified, the ordinary rules of algebra apply, with $\sqrt{2}$, $\sqrt{3}$, and so on treated like the pronumerals $x$, $y$.

• Adding and subtracting works only for like surds (the same number under the root). Simplify first so the like surds become visible, then add the coefficients: $\sqrt{50} + \sqrt{18} = 5\sqrt{2} + 3\sqrt{2} = 8\sqrt{2}$. Unlike surds, such as $\sqrt{2} + \sqrt{3}$, cannot be combined at all.
• Multiplying uses $\sqrt{a}\,\sqrt{b} = \sqrt{ab}$, then check whether the result simplifies further: $2\sqrt{6} \times 5\sqrt{10} = 10\sqrt{60} = 10 \times 2\sqrt{15} = 20\sqrt{15}$.
• Expanding brackets is exactly the algebra of the previous page, with the same three special expansions. In particular $(\sqrt{a} + \sqrt{b})(\sqrt{a} - \sqrt{b}) = a - b$, a difference of squares that turns two surds into a plain integer. For example $(\sqrt{7} + \sqrt{3})(\sqrt{7} - \sqrt{3}) = 7 - 3 = 4$. That single identity is the engine behind rationalising a binomial denominator.

Rationalising the denominator

It is conventional, and almost always required for full marks, to leave no surd in a denominator. Removing it is called rationalising, and there are two cases.

A single-term denominator (a surd or a multiple of one). Multiply the top and bottom by that surd. Because $\sqrt{a}\,\sqrt{a} = a$, the surd in the denominator becomes rational:

A two-term (binomial) denominator such as $\sqrt{5} + 1$ or $2\sqrt{3} - \sqrt{6}$. Multiply the top and bottom by the conjugate, which is the same expression with the middle sign reversed. The denominator then becomes a difference of squares and the surds vanish. The stepped figures below carry $\dfrac{4}{\sqrt{5} + 1}$ through the whole process.

Rationalise a binomial denominator, stage by stage

Stage 1, spot the conjugate. The denominator is $\sqrt{5} + 1$. Write down its conjugate, $\sqrt{5} - 1$ (same terms, opposite middle sign). Multiplying these two will give $(\sqrt{5})^2 - 1^2$, a difference of squares with no surd, so the plan is to multiply top and bottom by $\sqrt{5} - 1$.

Stage 2, multiply by $1$. Multiply the fraction by $\dfrac{\sqrt{5} - 1}{\sqrt{5} - 1}$. That quotient equals $1$, so the value of the fraction is unchanged; only its appearance changes. The numerator becomes $4(\sqrt{5} - 1)$ and the denominator becomes $(\sqrt{5} + 1)(\sqrt{5} - 1)$.

Stage 3, difference of squares. Expand the denominator with $(A + B)(A - B) = A^2 - B^2$. Here $A = \sqrt{5}$ and $B = 1$, so the denominator is $(\sqrt{5})^2 - 1^2 = 5 - 1 = 4$. The surd is gone: the denominator is now the integer $4$.

Stage 4, cancel and finish. The numerator $4(\sqrt{5} - 1)$ and the denominator $4$ share the factor $4$. Cancel it to reach the answer $\sqrt{5} - 1$, which has no surd in any denominator and so is fully rationalised.

The index laws (integer indices)

A power $a^n$ is repeated multiplication, and five laws let you combine powers of the same base. They are the rules you will use to differentiate, to handle exponentials, and to tidy almost any algebraic expression:

• Multiplying adds indices: $a^m \times a^n = a^{m+n}$.
• Dividing subtracts indices: $\dfrac{a^m}{a^n} = a^{m-n}$.
• Power of a power multiplies indices: $(a^m)^n = a^{mn}$.
• Power of a product distributes: $(ab)^n = a^n b^n$.
• Power of a quotient distributes: $\left(\dfrac{a}{b}\right)^n = \dfrac{a^n}{b^n}$.

When an expression has several factors, work through it systematically: the numbers first, then each pronumeral in turn. For instance $(20x^7y^3) \div (4x^5y^3) = 5x^{2}y^{0} = 5x^2$, since $20 \div 4 = 5$, $7 - 5 = 2$ and $3 - 3 = 0$ (and any non-zero base to the power $0$ is $1$). A common slip is to add the coefficients instead of multiplying them, or to apply a law across different bases: $a^3 \times b^2$ does not simplify, because the bases differ.

How exam questions ask about surds and indices

These skills underpin many longer questions, and a handful of command words tell you exactly which one is wanted:

• "Simplify" a surd means take out every square factor and collect like surds. State the answer in simplest surd form (no square factor left under the root, no like surds left uncombined).
• "Express in simplest exact form" or "leave your answer in surd form" is a signal not to give a decimal: keep $\sqrt{30}$, do not write $5.48$.
• "Rationalise the denominator" means clear the surd from the bottom: multiply by the surd for a single term, or by the conjugate for a two-term denominator, then use the difference of squares.
• "Show that" a surd expression equals a given value usually means rationalise or expand until both sides match; write every line so the marker can follow the manipulation.
• "Simplify, leaving your answer with positive indices" is an index-law question: combine the powers, then make sure no negative or zero index is left in the final form.
• Pythagoras or geometry problems often produce a surd answer ($d^2 = 30$ gives $d = \sqrt{30}$); take the positive root for a length, and simplify the surd.