ML Aggarwal Class 8 Solutions for ICSE Maths Chapter 3 Squares and Square Roots Ex 3.1
Question 1.
Which of the following natural numbers are perfect squares? Give reasons in support of your answer.
(i) 729
(ii) 5488
(iii) 1024
(iv) 243
Solution:
(i) 729
= 3 × 3 × 3 × 3 × 3 × 3
729 is the product of pairs of equal prime factors.
∴ 729 is a perfect square.
(ii) 5488
= 2 × 2 × 2 × 2 × 7 × 7 × 7
After pairing the same prime factors,
we see that one factor 7 is left unpaired.
So, 5488 is not a perfect square.
(iii) 1024
= 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2
After pairing the same prime factors,
we see that there is no factor left.
So, 1024 is a perfect square.
(iv) 243
= 3 × 3 × 3 × 3 ×3
After pairing the same prime factors.
We see that factor 3 is left unpaired.
So, 243 is not a perfect square.
Question 2.
Show that each of the following numbers is a perfect square. Also, find the number whose square is the given number.
(i) 1296
(ii) 1764
(iii) 3025
(iv) 3969
Solution:
(i) 1296
= 2 × 2 × 2 × 2 × 3 × 3 × 3 × 3
After pairing the same prime factors, we see that no factor is left.
So, 1296 is a perfect square and is the perfect square of 2 × 2 × 3 × 3 = 36
(ii) 1764
= 2 × 2 × 3 × 3 × 7 × 7
After pairing the same factors, no factor is left.
So, 1764 is a perfect square and
1764 is the perfect square of 2 × 3 × 7 = 42
(iii) 3025
= 5 × 5 × 11 × 11
After pairing the same prime factors, we see that no factor is left.
So, 3025 is a perfect square and is the perfect square of 5 × 11 = 55
(iv) 3969
= 3 × 3 × 3 × 3 × 7 × 7
After pairing the same prime factors, we see that no factor is left.
So, 3969 is a perfect square and is the square of 3 × 3 × 7 = 63
Question 3.
Find the smallest natural number by which 1008 should be multiplied to make it a perfect square.
Solution:
1008
= 2 × 2 × 2 × 2 × 3 × 3 × 7
After pairing the same kind of prime factor, one factor 7 is left.
So, by multiplying 1008 by 7
We shall get a perfect square
∴ Required smallest number = 7
Question 4.
Find the smallest natural number by which 5808 should be divided to make it a perfect square. Also, find the number whose square is the resulting number.
Solution:
5808
= 2 × 2 × 2 × 2 × 3 × 11 × 11
After pairing the same kind of prime factors,
we see that factor 3 is left.
So, by dividing the number by 3, we get a perfect square.
∴ The square root of the resulting number
= 2 × 2 × 11 = 44