Arithmetic

Lessons

Senior 4 · Basic Mathematics

Arithmetic

5 lessons

Lesson 3 of 5

Absolute value, powers and radicals

25 min

Distance without direction

A bus stands at the kilometre marker 37 on a main road and a motorbike at marker 52. They are $52 - 37 = 15$ km apart. If the motorbike were at marker 22 instead, they would be $37 - 22 = 15$ km apart too. The distance is always the larger marker minus the smaller one, whatever the direction. Mathematics has a short way to write "the size of a difference, ignoring its sign": the absolute value. In this lesson you also revise the laws of powers and learn to simplify square roots, three tools that appear in every later topic of this course, from interest to statistics.

Definition

Absolute value of a real number $x$ , written $∣ x ∣$ : its distance from 0 on the number line.

|x| = \begin{cases} x & \text{if } x \ge 0 \\ -x & \text{if } x \lt 0 \end{cases}

So $|x| \ge 0$ always, and $∣ a - b ∣$ is the distance between $a$ and $b$ .

Examples: $∣ 7 ∣ = 7$ , $∣ - 7 ∣ = 7$ , $∣ 0 ∣ = 0$ , $∣ 22 - 37 ∣ = ∣ - 15 ∣ = 15$ .

Properties and simple problems

Property	Example
$∣ - a ∣ = ∣ a ∣$	$∣ - 4 ∣ = ∣ 4 ∣ = 4$
$\|ab\| = \|a\|\,\|b\|$	$\|(-3)(5)\| = 15 = 3 \times 5$
$\left\|\frac{a}{b}\right\| = \frac{\|a\|}{\|b\|}$ , $b \ne 0$	$\left\|\frac{-12}{4}\right\| = \frac{12}{4} = 3$
$\|a + b\| \le \|a\| + \|b\|$	$\|3 + (-5)\| = 2 \le 8$

Two facts solve most problems, for $r \ge 0$ :

|x - a| = r \iff x = a - r \ \text{or}\ x = a + r, \qquad |x - a| \le r \iff a - r \le x \le a + r .

An oven for baking bread must stay within $5^\circ\mathrm{C}$ of $180^\circ\mathrm{C}$ : $|T - 180| \le 5$ , that is $175 \le T \le 185$ .

Worked example 1: equations and a tolerance

(a) Solve $∣ 2 x - 3 ∣ = 7$ . (b) A bag of flour is accepted if its mass $m$ (kg) satisfies $|m - 25| \le 0.4$ . Which of 24.5 kg, 25.3 kg and 25.4 kg are accepted?

Solution

$2 x - 3 = 7$ gives $x = 5$ ; $2 x - 3 = - 7$ gives $x = - 2$ . Check: $∣ 7 ∣ = 7$ and $∣ - 7 ∣ = 7$ . Solutions: $x = - 2$ or $x = 5$ .
The condition means $24.6 \le m \le 25.4$ . 24.5 kg is rejected ( $|24.5 - 25| = 0.5 \gt 0.4$ ); 25.3 kg and 25.4 kg are accepted.

Definition

Power: for a real number $a$ and a natural number $n$ , $a^n = a \times a \times \dots \times a$ ( $n$ factors). $a$ is the base and $n$ the exponent (index). For $a \ne 0$ : $a^{0} = 1$ and $a^{-n} = \frac{1}{a^n}$ .

Example: $2^{5} = 32$ , $5^{0} = 1$ , $10^{-3} = \frac{1}{1000} = 0.001$ .

Important

Laws of powers ( $a, b \ne 0$ ; $m, n$ integers):

a^m \times a^n = a^{m+n}, \qquad \frac{a^m}{a^n} = a^{m-n}, \qquad (a^m)^n = a^{mn}, \qquad (ab)^n = a^n b^n, \qquad \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n} .

Standard form (scientific notation) writes a number as $A \times 10^n$ with $1 \le A \lt 10$ : the 13 246 394 residents counted in the 2022 census are about $1.32 \times 10^7$ people.

Worked example 2: using the laws

Simplify (a) $\frac{(2x^3)^2 \times x^{-4}}{4x}$ (b) $\left(\frac{2}{3}\right)^{-2}$ (c) $(3 \times 10^4) \times (5 \times 10^{-7})$ in standard form.

Solution

$\frac{4x^6 \times x^{-4}}{4x} = \frac{4x^2}{4x} = x$ .
$\left(\frac{2}{3}\right)^{-2} = \left(\frac{3}{2}\right)^2 = \frac{9}{4}$ .
$15 \times 10^{-3} = 1.5 \times 10^{-2}$ .

Radicals

The square root of $a \ge 0$ , written $\sqrt{a}$ , is the non-negative number whose square is $a$ ; the $n$ th root $\sqrt[n]{a}$ is the number whose $n$ th power is $a$ . Roots are powers with fractional exponents: $\sqrt[n]{a} = a^{\frac{1}{n}}$ and $\sqrt[n]{a^m} = a^{\frac{m}{n}}$ . So $27^{\frac{2}{3}} = \left(\sqrt[3]{27}\right)^2 = 3^2 = 9$ .

For $a, b \ge 0$ :

\sqrt{ab} = \sqrt{a}\,\sqrt{b}, \qquad \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}\ (b \gt 0), \qquad \sqrt{a^2} = a .

To simplify a square root, take out the largest square factor: $\sqrt{72} = \sqrt{36 \times 2} = 6\sqrt{2}$ . Only like radicals (the same root of the same number) can be added: $5\sqrt{2} + 3\sqrt{2} = 8\sqrt{2}$ , but $\sqrt{2} + \sqrt{3}$ stays as it is.

Radicals turn up in real objects. An A4 sheet measures 210 mm by 297 mm, and $\frac{297}{210} \approx 1.414 \approx \sqrt{2}$ . The $\sqrt{2}$ shape is chosen so that cutting a sheet in half gives two smaller sheets of exactly the same shape:

Definition

Rationalising the denominator: rewriting a fraction so that no radical remains in the denominator.

For $\frac{k}{\sqrt{a}}$ , multiply top and bottom by $\sqrt{a}$ .
For $\frac{k}{p + \sqrt{q}}$ , multiply top and bottom by the conjugate $p - \sqrt{q}$ , using $(p + \sqrt{q})(p - \sqrt{q}) = p^2 - q$ .

Example: $\frac{6}{\sqrt{3}} = \frac{6\sqrt{3}}{3} = 2\sqrt{3}$ .

Worked example 3: simplify and rationalise

(a) Simplify $\sqrt{50} + \sqrt{18} - \sqrt{8}$ . (b) Rationalise $\frac{4}{3 - \sqrt{5}}$ . (c) Expand $(2 + \sqrt{3})^2$ .

Solution

$5\sqrt{2} + 3\sqrt{2} - 2\sqrt{2} = 6\sqrt{2}$ .
$\frac{4}{3 - \sqrt{5}} \times \frac{3 + \sqrt{5}}{3 + \sqrt{5}} = \frac{4(3 + \sqrt{5})}{9 - 5} = 3 + \sqrt{5}$ .
$4 + 4\sqrt{3} + 3 = 7 + 4\sqrt{3}$ .

Warning

Roots do not split over sums. $\sqrt{9 + 16} = \sqrt{25} = 5$ , but $\sqrt{9} + \sqrt{16} = 7$ . Likewise powers do not split over sums: $(a + b)^2 \ne a^2 + b^2$ . And a negative exponent does not make a number negative: $2^{-3} = \frac{1}{8}$ , not $- 8$ .

Tip

Learn the squares up to $15^{2} = 225$ and the cubes up to $5^{3} = 125$ . Then, to simplify $\sqrt{n}$ , test the square factors 4, 9, 16, 25, 36, 49, 64, … from the largest down, and stop at the first one that divides $n$ .

1st Class Maths: Rationalising the Denominator. Watch both cases: a single root on the bottom, and a bracket that needs its conjugate.

✦

Practice

Try each one before opening the answers. They get harder as you go.

Evaluate (a) $∣ - 9 ∣ + ∣ 4 - 10 ∣$ (b) $|3 - 8| \times |-2|$ .
Solve (a) $∣ x + 4 ∣ = 6$ (b) $|x - 12| \le 3$ .
Simplify (a) $x^5 \times x^{-2}$ (b) $\frac{(3a^2)^3}{9a^4}$ (c) $16^{\frac{3}{4}}$ (d) $8^{-\frac{1}{3}}$ .
Write in standard form: (a) 0.000 52 (b) $4.2 \times 10^3 \times 3 \times 10^5$ .
Simplify (a) $\sqrt{48}$ (b) $2\sqrt{12} + \sqrt{27}$ (c) $\sqrt{5} \times \sqrt{15}$ .
Rationalise (a) $\frac{10}{\sqrt{5}}$ (b) $\frac{2}{\sqrt{7} + \sqrt{3}}$ .
A machine fills 2-litre bottles; a bottle passes if $|V - 2000| \le 15$ , with $V$ in mL. Write the condition without absolute value, and decide whether 1984 mL and 2015 mL pass.

Show answersHide answers

(a) $9 + 6 = 15$ (b) $5 \times 2 = 10$ .
(a) $x = 2$ or $x = - 10$ (b) $9 \le x \le 15$ .
(a) $x^{3}$ (b) $\frac{27a^6}{9a^4} = 3a^2$ (c) $(\sqrt[4]{16})^3 = 8$ (d) $\frac{1}{2}$ .
(a) $5.2 \times 10^{-4}$ (b) $12.6 \times 10^8 = 1.26 \times 10^9$ .
(a) $4\sqrt{3}$ (b) $4\sqrt{3} + 3\sqrt{3} = 7\sqrt{3}$ (c) $\sqrt{75} = 5\sqrt{3}$ .
(a) $\frac{10\sqrt{5}}{5} = 2\sqrt{5}$ (b) $\frac{2(\sqrt{7} - \sqrt{3})}{7 - 3} = \frac{\sqrt{7} - \sqrt{3}}{2}$ .
$1985 \le V \le 2015$ . 1984 mL fails (it is 16 mL short); 2015 mL passes (exactly at the limit).

Quick check

Which values of x satisfy |x − 3| = 5?

Key points

$∣ x ∣$ is the distance from 0; $|x - a| \le r \iff a - r \le x \le a + r$ .
$a^m a^n = a^{m+n}$ , $(a^m)^n = a^{mn}$ , $a^{0} = 1$ , $a^{-n} = \frac{1}{a^n}$ , $a^{\frac{m}{n}} = \sqrt[n]{a^m}$ .
Simplify $\sqrt{n}$ by taking out the largest square factor; add only like radicals.
Rationalise with $\sqrt{a}$ or with the conjugate: $(p + \sqrt{q})(p - \sqrt{q}) = p^2 - q$ .

Sources and credits

Written by Ganzaa: the explanations, the diagrams, the worked examples, the practice items and the questions. The road, oven, flour and bottle figures are illustrative; the census total is from NISR's 2022 census.
Video: Rationalising the Denominator - GCSE Higher Maths, 1st Class Maths, embedded from YouTube.

Property	Example
$∣ - a ∣ = ∣ a ∣$	$∣ - 4 ∣ = ∣ 4 ∣ = 4$
$\|ab\| = \|a\|\,\|b\|$	$\|(-3)(5)\| = 15 = 3 \times 5$
$\left\|\frac{a}{b}\right\| = \frac{\|a\|}{\|b\|}$ , $b \ne 0$	$\left\|\frac{-12}{4}\right\| = \frac{12}{4} = 3$
$\|a + b\| \le \|a\| + \|b\|$	$\|3 + (-5)\| = 2 \le 8$

1Distance without direction

2Properties and simple problems

3Radicals

Distance without direction

Properties and simple problems

Radicals