GSEB Class 9 Maths Solutions Chapter 1 Number Systems Exercise 1.6

Question 1. Find
(i) \( 64^{\frac{1}{2}} \)
(ii) \( 32^{\frac{1}{5}} \)
(iii) \( 125^{\frac{1}{3}} \)
Answer:
(i) \( 64^{\frac{1}{2}} = \left(8^{2}\right)^{\frac{1}{2}} = 8^{2 \times \frac{1}{2}} = 8^1 = 8 \)
(ii) \( 32^{\frac{1}{5}} = \left(2^{5}\right)^{\frac{1}{5}} = 2^{5 \times \frac{1}{5}} = 2^1 = 2 \)
(iii) \( 125^{\frac{1}{3}} = \left(5^{3}\right)^{\frac{1}{3}} = 5^{3 \times \frac{1}{3}} = 5^1 = 5 \)
In simple words: To find these values, first write the base number as a power, then use the exponent rule \( (a^m)^n = a^{m \times n} \). This simplifies the calculation and helps you find the final result.

Question 2. Find
(i) \( 9^{\frac{3}{2}} \)
(ii) \( 32^{\frac{2}{5}} \)
(iii) \( 16^{\frac{3}{4}} \)
(iv) \( 125^{-\frac{1}{3}} \)
Answer:
(i) \( 9^{\frac{3}{2}} = \left(3^{2}\right)^{\frac{3}{2}} = 3^{2 \times \frac{3}{2}} = 3^3 = 3 \times 3 \times 3 = 27 \)
(ii) \( 32^{\frac{2}{5}} = \left(2^{5}\right)^{\frac{2}{5}} = 2^{5 \times \frac{2}{5}} = 2^2 = 2 \times 2 = 4 \)
(iii) \( 16^{\frac{3}{4}} = \left(2^{4}\right)^{\frac{3}{4}} = 2^{4 \times \frac{3}{4}} = 2^3 = 8 \)
(iv) \( 125^{-\frac{1}{3}} = \left(5^{3}\right)^{-\frac{1}{3}} = 5^{3 \times (-\frac{1}{3})} = 5^{-1} = \frac{1}{5} \)
In simple words: First, convert the base number into a power. Then, apply the power rule \( (a^m)^n = a^{m \times n} \) to multiply the exponents. Simplify the result to find your answer. For negative exponents, remember that \( a^{-n} = \frac{1}{a^n} \).

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) \( 7^{\frac{1}{2}} \cdot 8^{\frac{1}{2}} \)
Answer:
(i) \( 2^{\frac{2}{3}} \cdot 2^{\frac{1}{5}} = 2^{\frac{2}{3} + \frac{1}{5}} = 2^{\frac{2 \times 5 + 3 \times 1}{15}} = 2^{\frac{10+3}{15}} = 2^{\frac{13}{15}} \)
(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) \( 7^{\frac{1}{2}} \cdot 8^{\frac{1}{2}} = (7 \times 8)^{\frac{1}{2}} \), using the rule \( a^m \times b^m = (a \times b)^m \)
\( = 56^{\frac{1}{2}} \)
In simple words: When multiplying powers with the same base, you add the exponents. For a power of a fraction, apply the exponent to both the numerator and denominator. When dividing powers with the same base, you subtract the exponents. When multiplying powers with different bases but the same exponent, you multiply the bases and keep the exponent.