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A Square and A Cube – Class 8 Mathematics notes, important questions and MCQs

NCERT Class 8 Mathematics, chapter 1. Notes, key terms, important questions with answers and MCQs.

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Chapter overview

A square number is n × n and a cube number is n × n × n. Their patterns help find square roots and cube roots quickly.

Squares and square roots

Perfect squares end only in 0, 1, 4, 5, 6 or 9.
Sum of the first n odd numbers = n².
Square root by prime factorisation: pair the prime factors (√144 = √(2⁴ × 3²) = 2² × 3 = 12).

Cubes and cube roots

Cubes: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000.
Cube root by prime factorisation: make triplets (∛1728 = 2² × 3 = 12).
Unit digit of a cube: 2 → 8, 3 → 7, 7 → 3, 8 → 2; others stay the same.

Key terms

Perfect square: a number that is the square of a whole number
Perfect cube: a number that is the cube of a whole number
Square root: a number that gives the original number when multiplied by itself, e.g. √49 = 7

Formulas

n² = sum of first n odd numbers
(a + b)² shortcut: 35² = 3 × 4 | 25 = 1225

Important questions with answers

Is 2,352 a perfect square? Give a reason from its last digit or factors.2 marks

2,352 = 2⁴ × 3 × 7²; 3 appears once, so it is not a perfect square.

Find the cube root of 13,824 by prime factorisation.3 marks

13,824 = 2⁹ × 3³ = (2³ × 3)³ = 24³. Cube root = 24.

Find the smallest number by which 1,323 must be divided to make it a perfect cube, and find that cube root. Also find the square root of 6,084.5 marks

1,323 = 3³ × 7². 7² is not in a group of three, so divide by 49: 1,323 ÷ 49 = 27 = 3³, cube root 3. 6,084 = 2² × 3² × 13², so √6,084 = 2 × 3 × 13 = 78.

Quick revision

√ uses pairs; ∛ uses triplets.
12² = 144, 15² = 225, 20² = 400.
10³ = 1000, 12³ = 1728.

Exam tips

A number ending in 2, 3, 7 or 8 is never a perfect square.
To make a number a perfect cube, multiply by the primes needed to complete triplets.

Practice MCQs

Which is a perfect square?
1. 48
2. 64
3. 72
4. 98
Show answer
Answer: (b) 64. 8 × 8.
The cube of 5 is
1. 25
2. 125
3. 15
4. 75
Show answer
Answer: (b) 125. 5 × 5 × 5.
√196 is
1. 13
2. 16
3. 14
4. 12
Show answer
Answer: (c) 14. 196 = 2² × 7², root = 14.
The smallest number by which 72 must be multiplied to make it a perfect square is
1. 2
2. 3
3. 6
4. 8
Show answer
Answer: (a) 2. 72 = 2³ × 3²; one more 2 needed.
√1764 equals
1. 38
2. 42
3. 44
4. 48
Show answer
Answer: (b) 42. 42 × 42 = 1764.
∛2744 is
1. 12
2. 14
3. 16
4. 18
Show answer
Answer: (b) 14. 2744 = 2³ × 7³.
The smallest number by which 1350 must be multiplied to make a perfect cube is
1. 20
2. 10
3. 5
4. 4
Show answer
Answer: (a) 20. 1350 = 2 × 3³ × 5²; need 2² × 5 = 20.
The smallest number by which 72 must be multiplied to make it a perfect square is
1. 3
2. 2
3. 6
4. 4
Show answer
Answer: (b) 2. 72 = 2³ × 3²; one more 2.

All chapters of Class 8 Mathematics

Written by Kundali IQ from standard, well-established facts to help with revision. This is not an official NCERT publication; always read your textbook too.