MathsICSE Class 9 · Rational and Irrational Numbers

Surds: Simplifying, Rationalising the Denominator and Representing √2, √3, √5 on the Number Line (ICSE Class 9)

Updated 17 October 2026 · Fact-checked

A surd is an irrational root such as √2 or ∛5. Simplify √(a²b) as a√b, add or subtract only like surds, and rationalise a denominator by multiplying by its conjugate: (a + √b)(a − √b) = a² − b. Square roots like √2, √3, √5 are drawn on the number line using right triangles.

Each hypotenuse is swung down onto the number line with a compass, so √2 and √3 land exactly between 1 and 2.

Illustration drawn for QuestionBuddy.

Understand Surds: Simplification, Rationalising the Denominator and the Number Line

Surds and simplification. A surd is an irrational root. To simplify, pull out perfect squares: $72 = 36 \times 2 = 62$ . Laws: $a b = ab$ and $\frac{a}{b} = \frac{a}{b}$ . Like surds (same radical) can be added: $50 + 18 - 8 = 52 + 32 - 22 = 62$ . $a + b \neq = a + b$ .

Rationalising. To remove a surd from the denominator:

single surd: $\frac{1}{a} = \frac{a}{a}$ ;
binomial: multiply numerator and denominator by the conjugate (change the sign between the terms), because $(p + q) (p - q) = p^{2} - q$ and $(a + b) (a - b) = a - b$ .

*Examples.* $\frac{1}{3 + 2} = \frac{3 - 2}{9 - 2} = \frac{3 - 2}{7}$ ; $\frac{5}{7 - 2} = \frac{5 ( 7 + 2 )}{5} = 7 + 2$ ; $\frac{3 + 2}{3 - 2} = (3 + 2)^{2} = 5 + 26$ .

**Finding $a$ and $b$ .** If $\frac{2 + 3}{2 - 3} = a + b 3$ , rationalise: $(2 + 3)^{2} = 7 + 43$ , so $a = 7$ , $b = 4$ (compare the rational and irrational parts).

Reciprocal pairs. If $x = 3 + 22$ then $\frac{1}{x} = 3 - 22$ (since $9 - 8 = 1$ ), so $x + \frac{1}{x} = 6$ and $x^{2} + \frac{1}{x ^{2}} = 6^{2} - 2 = 34$ .

Number line. $2$ is the hypotenuse of a right triangle with legs $1, 1$ : with centre $O$ and radius equal to this hypotenuse, an arc marks $2$ on the line. $3$ comes from legs $2, 1$ , and $5$ from legs $2, 1$ ( $4 + 1$ ). Each new surd uses the previous length as one leg ('square-root spiral').

Level. Class IX is a school-exam class. Conjugate multiplication is routine; finding $a$ and $b$ and the reciprocal-pair values are at the hardest school-exam level.

Surds: Simplification, Rationalising the Denominator and the Number Line: formulas to remember

Simplify: $a^{2} b = a b$; Take out squares.
Conjugate product: $(p + q) (p - q) = p^{2} - q$; Removes the surd.
Surds product: $a b = ab$; Non-negative a, b.
Reciprocal pair: $(p + q)^{- 1} = \frac{p - q}{p ^{2} - q}$; When p² − q = 1 it is just the conjugate.

How to solve Surds: Simplification, Rationalising the Denominator and the Number Line questions

Simplify each surd first, then rationalise, then compare rational and irrational parts.

1Write each surd in simplest form by taking out perfect squares.
2Combine like surds.
3To rationalise, multiply top and bottom by the conjugate of the denominator.
4Expand using $a^{2} - b^{2}$ in the denominator and simplify.
5Compare rational and irrational parts to find $a$ , $b$ .
6Check numerically with decimals ( $2 \approx 1.414$ ).

Surds: Simplification, Rationalising the Denominator and the Number Line in real life

Each concept, met in an everyday situation. Read the situation, try the problem, then check the solution. The one-line hook is what to remember.

Diagonal of a square plate

Suppose a square tile has side $1$ unit and a designer needs the diagonal length for a border strip.

Give its exact length and mark it on a number line.

Try it first, then show the solutionSolution

Diagonal $= 1^{2} + 1^{2} = 2$ .
On the line, draw a unit square on $[0, 1]$ ; swing an arc of radius $2$ from $O$ to mark $2 \approx 1.414$ .

Answer: $2$ units.

Memory hook: The diagonal is the hypotenuse of a 1-1 right triangle.

Exact answers without a calculator

Imagine a worksheet answer $\frac{1}{3 + 2}$ that must be written without a surd in the denominator.

Rationalise and find a decimal using $2 \approx 1.414$ .

Try it first, then show the solutionSolution

$\frac{3 - 2}{7}$ .
$\approx \frac{1.586}{7} = 0.2266$ .

Answer: $\frac{3 - 2}{7} \approx 0.2266$ .

Memory hook: Conjugate removes the surd.

Tricks to remember Surds: Simplification, Rationalising the Denominator and the Number Line

Short memory aids. Say each one aloud once, then use it on a question the same day.

Flip the sign
The conjugate of p + √q is p − √q.
Use it when: Rationalising.
Like with like
Only surds with the same radical add.
Use it when: Combining terms.
Difference of squares
Denominator becomes a rational number.
Use it when: Why it works.

Surds: Simplification, Rationalising the Denominator and the Number Line: solved examples

Ordered from board level to HOTS. Try each one before you open the solution.

Example 1: Simplify · 1 mark · MCQBoard level

$72$ in simplest form is:
(a) $62$
(b) $26$
(c) $123$
(d) $98$

Try it first, then show the solutionSolution

$72 = 36 \times 2 = 62$ .

Answer: (a) $62$

Example 2: Like surds · 3 marks · Short answerBoard level

Simplify $50 + 18 - 8$ .

Try it first, then show the solutionSolution

$52 + 32 - 22$ .
$= 62$ .

Answer: $62$

Example 3: Rationalising · 4 marks · Short answerHarder board level

Rationalise (i) $\frac{1}{3 + 2}$ (ii) $\frac{5}{7 - 2}$ .

Try it first, then show the solutionSolution

(i) $\frac{3 - 2}{9 - 2} = \frac{3 - 2}{7}$ .
(ii) $\frac{5 ( 7 + 2 )}{7 - 2} = 7 + 2$ .

Answer: (i) $\frac{3 - 2}{7}$ (ii) $7 + 2$

Example 4: A ratio of surds · 4 marks · Short answerHarder board level

Simplify $\frac{3 + 2}{3 - 2}$ and find its approximate value ( $6 \approx 2.449$ ).

Try it first, then show the solutionSolution

Multiply by the conjugate: $\frac{( 3 + 2 ) ^{2}}{3 - 2} = 3 + 2 + 26$ .
$= 5 + 26 \approx 5 + 4.898 = 9.898$ .

Answer: $5 + 26 \approx 9.898$

Example 5: A reciprocal pair · 5 marks · Long answerHOTS / competency

If $x = 3 + 22$ , find $\frac{1}{x}$ , $x + \frac{1}{x}$ and $x^{2} + \frac{1}{x ^{2}}$ .

Try it first, then show the solutionSolution

$\frac{1}{x} = \frac{3 - 2 2}{9 - 8} = 3 - 22$ .
$x + \frac{1}{x} = 6$ .
$x^{2} + \frac{1}{x ^{2}} = (x + \frac{1}{x})^{2} - 2 = 36 - 2 = 34$ .
Check: $x^{2} = 17 + 122$ and $\frac{1}{x ^{2}} = 17 - 122$ , sum $34$ . ✓

Answer: $3 - 22$ ; $6$ ; $34$

Where students lose marks in Surds: Simplification, Rationalising the Denominator and the Number Line

The slips examiners see most often, and what to write instead.

Mistake: Adding unlike surds
Only same radicals combine.
Wrong working
$2 + 3 = 5$ .
Right working
Cannot be combined: $\approx 3.146$ .
Do this: Check the radicals.
Usually costs: Whole answer.
Mistake: Wrong conjugate
Change only the sign between the terms.
Wrong working
Conjugate of $3 + 2$ is $- 3 - 2$ .
Right working
$3 - 2$ .
Do this: Flip one sign.
Usually costs: Whole answer.
Mistake: Forgetting to multiply the numerator
Both top and bottom are multiplied.
Wrong working
$\frac{1}{3 + 2} = \frac{1}{7}$ .
Right working
$\frac{3 - 2}{7}$ .
Do this: Multiply both.
Usually costs: Whole answer.

Exam tips

Class IX is examined in school papers; always simplify surds before rationalising.
Give exact answers unless a decimal is asked for.
In a and b problems, state that you are comparing rational and irrational parts.
Check with decimals when in doubt.

Practice questions on Surds: Simplification, Rationalising the Denominator and the Number Line

Write each answer on paper first. Open the hint only if you are stuck, and the answer only when you have finished.

Q1 · 1 mark · MCQBoard level
The simplest form of $48$ is:
(a) $212$
(b) $43$
(c) $34$
(d) $62$
Hint
$48 = 16 \times 3$ .
Answer
(b) $43$
Q2 · 3 marks · Short answerHarder board level
Rationalise $\frac{3}{5 + 2}$ .
Hint
Conjugate.
Answer
$\frac{3 ( 5 - 2 )}{5 - 2} = 5 - 2$ .
Q3 · 4 marks · Short answerHarder board level
Find rational numbers $a$ and $b$ if $\frac{2 + 3}{2 - 3} = a + b 3$ .
Hint
Rationalise and compare.
Answer
$\frac{( 2 + 3 ) ^{2}}{4 - 3} = 7 + 43$ , so $a = 7$ , $b = 4$ .
Q4 · 1 mark · Assertion-reasonHarder board level
Assertion (A): $2 + 3 = 5$ .
Reason (R): $a + b \neq = a + b$ in general.
(a) Both (A) and (R) are true and (R) explains (A).
(b) Both are true but (R) does not explain (A).
(c) (A) is true, (R) is false.
(d) (A) is false, (R) is true.
Hint
$1.414 + 1.732 = 3.146$ but $5 \approx 2.236$ .
Answer
(d)
Q5 · 4 marks · Case-basedHOTS / competency
Case study: On a number line, a student draws a right triangle with legs $1$ and $1$ to mark $2$ , then uses it to mark other surds.
(a) What is the hypotenuse of a triangle with legs $1$ and $1$ ?
(b) What legs give $5$ ?
(c) What legs give $3$ using the first length?
(d) Give the value of $5$ to three decimal places.
Hint
$5 = 2^{2} + 1^{2}$ .
Answer
(a) $2$ (b) $2$ and $1$ (c) $2$ and $1$ (d) $2.236$

Test your recall

Answer in your head or on paper first, then open the card. Pulling an answer out of memory makes it stick far better than re-reading.

1.Simplify √72.Show answer
6√2.
2.Conjugate of √7 − √2?Show answer
√7 + √2.
3.(p + √q)(p − √q) equals?Show answer
p² − q.
4.Which legs give √5?Show answer
2 and 1.

Surds: Simplification, Rationalising the Denominator and the Number Line: frequently asked questions

Do I use tables for √2?

No; values such as √2 ≈ 1.414 and √3 ≈ 1.732 are given or computed.

Does the syllabus ask for the number line?

Yes, the chapter lists representing √2, √3 and √5 on the number line.

What if the denominator is a single surd?

Multiply top and bottom by that surd.