Question

By what least number should the following numbers be multiplied so that they become perfect squares? Also
find the numbers whose squares the new numbers are.
(i) 171
(ii) 2880
(iii) 3332
(iv) 18225
(v) 4220​

Answer 1

I)

171= 3×3×19

ans - 19

( make their exponents even.)

ii)

Answer 2

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Page No 42:

Question 1:

Using the prime factorisation method, find which of the following numbers are perfect squares:
(i) 441
(ii) 576
(iii) 11025
(iv) 1176
(v) 5625
(vi) 9075
(vii) 4225
(viii) 1089

ANSWER:

A perfect square can always be expressed as a product of equal factors.

(i)
Resolving into prime factors:
441=49×9=7×7×3×3=7×3×7×3=21×21=(21)2


Thus, 441 is a perfect square.

(ii)
Resolving into prime factors:
576=64×9=8×8×3×3=2×2×2×2×2×2×3×3=24×24=(24)2


Thus, 576 is a perfect square.

(iii)
Resolving into prime factors:
11025=441×25=49×9×5×5=7×7×3×3×5×5=7×5×3×7×5×3=105×105=(105)2


Thus, 11025 is a perfect square.

(iv)
Resolving into prime factors:
1176=7×168=7×21×8=7×7×3×2×2×2


1176 cannot be expressed as a product of two equal numbers. Thus, 1176 is not a perfect square.

(v)
Resolving into prime factors:
5625=225×25=9×25×25=3×3×5×5×5×5=3×5×5×3×5×5=75×75=(75)2


Thus, 5625 is a perfect square.

(vi)
Resolving into prime factors:
9075=25×363=5×5×3×11×11=55×55×3


9075 is not a product of two equal numbers. Thus, 9075 is not a perfect square.

(vii)
Resolving into prime factors:
4225=25×169=5×5×13×13=5×13×5×13=65×65=(65)2


Thus, 4225 is a perfect square.

(viii)
Resolving into prime factors:
1089=9×121=3×3×11×11=3×11×3×11=33×33=(33)2


Thus, 1089 is a perfect square.

Page No 42:

Question 2:

Show that each of the following numbers is a perfect square. In each case, find the number whose square is the given number:
(i) 1225
(ii) 2601
(iii) 5929
(iv) 7056
(v) 8281

ANSWER:

A perfect square is a product of two perfectly equal numbers.

(i)
Resolving into prime factors:

1225=25×49=5×5×7×7=5×7×5×7=35×35=(35)2


Thus, 1225 is the perfect square of 35.

(ii)
Resolving into prime factors:
2601=9×289=3×3×17×17=3×17×3×17=51×51=(51)2


Thus, 2601 is the perfect square of 51.

(iii)
Resolving into prime factors:

5929=11×539=11×7×77=11×7×11×7=77×77=(77)2


Thus, 5929 is the perfect square of 77.

(iv)
Resolving into prime factors:
7056=12×588=12×7×84=12×7×12×7=(12×7)2=(84)2


Thus, 7056 is the perfect square of 84.

(v)
Resolving into prime factors:

8281=49×169=7×7×13×13=7×13×7×13=(7×13)2=(91)2


Thus, 8281 is the perfect square of 91.

Page No 42:

Question 3:

By what least number should the given number be multiplied to get a perfect square number? In each case, find the number whose square is the new number.
(i) 3975
(ii) 2156
(iii) 3332
(iv) 2925
(v) 9075
(vi) 7623
(vii) 3380
(viii) 2475