GSEB Solutions Class 9 Maths Chapter 1 Number Systems Ex 1.6

Gujarat Board GSEB Solutions Class 9 Maths Chapter 1 Number Systems Ex 1.6 Textbook Questions and Answers.

Gujarat Board Textbook Solutions Class 9 Maths Chapter 1 Number Systems Ex 1.6

Question 1.
Find
(i) 64\(^{\frac{1}{2}}\)
(ii) 32\(^{\frac{1}{5}}\)
(iii) 125\(^{\frac{1}{3}}\)
Solution:
(i) 64\(^{\frac{1}{2}}\) = \(\left(8^{2}\right)^{\frac{1}{2}}\) = 8\(^{2 \times \frac{1}{2}}\) = 8¹ = 8

(ii) 32\(^{\frac{1}{5}}\) = \(\left(2^{5}\right)^{\frac{1}{5}}\) = 2\(^{5 \times \frac{1}{5}}\) = 2¹ = 2

(iii) 125\(^{\frac{1}{3}}\) = \(\left(5^{3}\right)^{\frac{1}{3}}\) = 5\(^{3 \times \frac{1}{3}}\) = 5¹ = 5

Question 2.
Find
(i) 9\(^{\frac{3}{2}}\)
(ii) 32\(^{\frac{2}{5}}\)
(iii) 16\(^{\frac{3}{4}}\)
(iv) 125\(^{-\frac{1}{3}}\)
Solution:
(i) 9\(^{\frac{3}{2}}\) = \(\left(3^{2}\right)^{\frac{3}{2}}\) = 3\(^{2 \times \frac{3}{2}}\) = 3³ = 3 x 3 x 3 = 27

(ii) 32\(^{\frac{2}{5}}\) = \(\left(2^{5}\right)^{\frac{2}{5}}\) = 3\(^{5 \times \frac{2}{5}}\) = 2² = 2 x 2 = 4

(iii) 16\(^{\frac{3}{4}}\) = \(\left(2^{4}\right)^{\frac{3}{4}}\) = 2\(^{4 \times \frac{3}{4}}\) = 2³ = 8

(iv) 125\(^{-\frac{1}{3}}\) = (5³)^{\(\frac { 1 }{ 3 }\)} = \(5^{3 \times-\frac{3}{4}}\) = 5^-1 = \(\frac { 1 }{ 5 }\)

Question 3.
Simplify:
(i) \(2^{\frac{2}{3}} \cdot 2^{\frac{1}{5}}\)
(ii) \(\left(\frac{1}{3^{3}}\right)^{7}\)
(iii) \(\frac{11^{\frac{1}{2}}}{11^{\frac{1}{4}}}\)
(iv) \(7^{\frac{1}{2}} \cdot 8^{\frac{1}{2}}\)
Solution:
(i) \(2^{\frac{2}{3}} \cdot 2^{\frac{1}{5}}\) = \(2^{\frac{2}{3}} + 2^{\frac{1}{5}}\) = 2^{\(\frac { 2×5+3×1 }{ 15 }\)} = 2^{\(\frac { 10+3 }{ 15 }\)} = 2^{\(\frac { 13 }{ 15 }\)}

(ii) \(\left(\frac{1}{3^{3}}\right)^{7}\) = \(\frac{1}{\left(3^{3}\right)^{7}}\) = \(\frac{1}{3^{21}}\) = 3^-21

(iii) \(\frac{11^{\frac{1}{2}}}{11^{\frac{1}{4}}}\) = 11^{\(\frac { 1 }{ 2 }\) – \(\frac { 1 }{ 4 }\)} = 11^{\(\frac { 1 }{ 4 }\)}

(iv) \(7^{\frac{1}{2}} \cdot 8^{\frac{1}{2}}\) = (7 x 8)^{\(\frac { 1 }{ 2 }\)} [a^m x b^m = (a x b)^m]
= 56^{\(\frac { 1 }{ 2 }\)}