CBSE • Class 9 • Mathematics • Chapter 1 (Number Systems) • Exercise 1.4

Exercise 1.4: Number Systems — NCERT Solutions

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

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What this exercise covers

Rational exponentsLaws of exponentsSimplifying powers

Step-by-step solutions — Exercise 1.4

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

Exercise 1.4 Q1 • 3 marks

Evaluate using laws of exponents: (i) 22/3⋅21/32^{2/3} \cdot 2^{1/3} (ii) (31/2)4\left(3^{1/2}\right)^4 (iii) 71/3÷71/47^{1/3} \div 7^{1/4}.
Hint (Socratic — try this first)▾
Which exponent law applies to multiplying, raising a power to a power, and dividing powers with the same base?
Step-by-step solution▾

(i) am⋅an=am+na^m \cdot a^n = a^{m+n}: 22/3⋅21/3=223+13=21=22^{2/3} \cdot 2^{1/3} = 2^{\frac{2}{3}+\frac{1}{3}} = 2^{1} = 2

(ii) (am)n=amn(a^m)^n = a^{mn}: (31/2)4=312×4=32=9\left(3^{1/2}\right)^4 = 3^{\frac{1}{2}\times 4} = 3^{2} = 9

(iii) am÷an=am−na^m \div a^n = a^{m-n}: 71/3÷71/4=713−14=74−312=71/127^{1/3} \div 7^{1/4} = 7^{\frac{1}{3}-\frac{1}{4}} = 7^{\frac{4-3}{12}} = 7^{1/12}

Conclusion: (i) 22, (ii) 99, (iii) 71/127^{1/12}.

Common mistake:
Students multiply the exponents when they should add them (in part i), giving 22/92^{2/9} instead of 212^1.
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How to approach Exercise 1.4

  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 1.3

Real numbers — operations and representability; approximating irrational numbers.

Exercise 1.5 →

Rationalising denominators with surds and revisiting identities with roots.

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