CBSE • Class 9 • Mathematics • Chapter 1 (Number Systems) • Exercise 1.5

Exercise 1.5: Number Systems — NCERT Solutions

Rationalising denominators with surds and revisiting identities with roots.

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What this exercise covers

Rationalising factorSurd manipulationIdentities

Step-by-step solutions — Exercise 1.5

2 solved questions • Each solution includes a Socratic hint, full working and a common-mistake callout • Last reviewed 2026-09-03

Exercise 1.5 Q1 • 3 marks

Rationalise the denominator of 17−6\dfrac{1}{\sqrt{7} - \sqrt{6}}.
Hint (Socratic — try this first)▾
What expression should you multiply by so the denominator becomes a difference of squares with no surds?
Step-by-step solution▾

Conjugate method: Multiply numerator and denominator by the conjugate (7+6)(\sqrt{7} + \sqrt{6}):

17−6×7+67+6\frac{1}{\sqrt{7}-\sqrt{6}} \times \frac{\sqrt{7}+\sqrt{6}}{\sqrt{7}+\sqrt{6}}

Denominator (using (a−b)(a+b)=a2−b2(a-b)(a+b)=a^2-b^2): (7)2−(6)2=7−6=1(\sqrt{7})^2 - (\sqrt{6})^2 = 7 - 6 = 1

Result: =7+61=7+6= \frac{\sqrt{7}+\sqrt{6}}{1} = \sqrt{7} + \sqrt{6}

Conclusion: 17−6=7+6\dfrac{1}{\sqrt{7}-\sqrt{6}} = \sqrt{7} + \sqrt{6}.

Common mistake:
Students multiply by (7−6)(\sqrt7-\sqrt6) instead of the conjugate (7+6)(\sqrt7+\sqrt6), leaving a surd in the denominator.
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Exercise 1.5 Q2 • 2 marks

Rationalise the denominator of 423\dfrac{4}{2\sqrt{3}} and simplify.
Hint (Socratic — try this first)▾
What single surd can you multiply top and bottom by to clear the square root from the denominator?
Step-by-step solution▾

Multiply numerator and denominator by 3\sqrt{3}:

423×33=432×3=436\frac{4}{2\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{4\sqrt{3}}{2 \times 3} = \frac{4\sqrt{3}}{6}

Simplify the fraction: =233= \frac{2\sqrt{3}}{3}

Conclusion: 423=233\dfrac{4}{2\sqrt{3}} = \dfrac{2\sqrt{3}}{3}.

Common mistake:
Students forget to reduce 436\frac{4\sqrt3}{6} to lowest terms, or multiply by 232\sqrt3 unnecessarily.
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How to approach Exercise 1.5

  1. Re-read the chapter summary first. Open Number Systems and refresh the key concepts: Rational numbers, Irrational numbers, Decimal expansions, Laws of exponents.
  2. Try each problem yourself before opening the solution. Ten focused minutes per problem usually beats reading two finished solutions.
  3. Use Guru AI for the questions you get stuck on. The Socratic AI tutor walks you through it with hints instead of dictating the answer.
  4. Mark the questions you got wrong and revisit them after 24 hours — the spacing is what locks the method into long-term memory.

All exercises in Number Systems

  1. Exercise 1.1 — Rational vs irrational numbers — closure, density on the line, and locating simple surds.
  2. Exercise 1.2 — Decimal expansions of rationals — terminating and non-terminating recurring forms.
  3. Exercise 1.3 — Real numbers — operations and representability; approximating irrational numbers.
  4. Exercise 1.4 — Laws of exponents extended to rational and real bases and exponents.
  5. Exercise 1.5 — Rationalising denominators with surds and revisiting identities with roots.
  6. Exercise 1.6 — Representing irrational numbers like √x on the number line using constructions.

← Exercise 1.4

Laws of exponents extended to rational and real bases and exponents.

Exercise 1.6 →

Representing irrational numbers like √x on the number line using constructions.

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