Number Systems Exemplar — Class 9 Mathematics NCERT Solutions (Free)

Free step-by-step NCERT solutions for Class 9 Mathematics chapter "Number Systems Exemplar" — 7 important questions with detailed answers for CBSE board exam preparation.

TL;DR: Free step-by-step NCERT solutions for Class 9 Mathematics chapter "Number Systems Exemplar" — 7 important questions with detailed answers for CBSE boa…

By Syllab.in · Updated Jun 17, 2026

Key Questions Covered:

Show that 5 - 2√3 is irrational.
Find the value of (81)^(3/4).
Simplify: (√5 + √3)/(√5 - √3).
If x = 3 + 2√2, find x + 1/x.
Express 0.3456̄ (where 3456 repeats) as a fraction.
Prove that √2 is irrational.
+ 1 more questions in the full chapter

Solutions Summary:

Question	Status
Show that 5 - 2√3 is irrational.	✓ Solved
Find the value of (81)^(3/4).	✓ Solved
Simplify: (√5 + √3)/(√5 - √3).	✓ Solved
If x = 3 + 2√2, find x + 1/x.	✓ Solved
Express 0.3456̄ (where 3456 repeats) as a fraction.	✓ Solved
Prove that √2 is irrational.	✓ Solved

Showing 6 of 7 questions

Q1: Show that 5 - 2√3 is irrational.

Assume 5 - 2√3 is rational. Then 5 - 2√3 = p/q where p, q are integers and q ≠ 0. Rearranging: 2√3 = 5 - p/q = (5q - p)/q. So √3 = (5q - p)/(2q). Since p, q are integers, (5q - p)/(2q) is rational. But this means √3 is rational, which contradicts the fact that √3 is irrational. Therefore, 5 - 2√3 must be irrational.

Q2: Find the value of (81)^(3/4).

(81)^(3/4) = (81^(1/4))^3. Now 81 = 3^4, so 81^(1/4) = (3^4)^(1/4) = 3. Therefore (81)^(3/4) = 3^3 = 27.

Q3: Simplify: (√5 + √3)/(√5 - √3).

Multiply numerator and denominator by (√5 + √3): [(√5 + √3)/(√5 - √3)] × [(√5 + √3)/(√5 + √3)] = (√5 + √3)^2 / [(√5)^2 - (√3)^2] = (5 + 2√15 + 3) / (5 - 3) = (8 + 2√15) / 2 = 4 + √15.

Q4: If x = 3 + 2√2, find x + 1/x.

First find 1/x = 1/(3 + 2√2). Rationalize: 1/(3 + 2√2) × (3 - 2√2)/(3 - 2√2) = (3 - 2√2)/(9 - 8) = 3 - 2√2. Therefore x + 1/x = (3 + 2√2) + (3 - 2√2) = 6.

Q5: Express 0.3456̄ (where 3456 repeats) as a fraction.

Let x = 0.34563456... Then 10000x = 3456.34563456... Subtracting: 10000x - x = 3456. So 9999x = 3456, giving x = 3456/9999. Simplifying by GCD(3456, 9999) = 3: x = 1152/3333 = 384/1111.

Q6: Prove that √2 is irrational.

Assume √2 is rational. Then √2 = p/q in lowest terms (GCD(p,q) = 1). Squaring: 2 = p^2/q^2, so p^2 = 2q^2. This means p^2 is even, so p is even. Let p = 2m. Then (2m)^2 = 2q^2, so 4m^2 = 2q^2, giving 2m^2 = q^2. Thus q^2 is even, so q is even. But if both p and q are even, GCD(p,q) ≥ 2, contradicting our assumption. Therefore √2 is irrational.

Showing 6 of 7 questions. Visit the full page for complete solutions.