6 (a) Simplify \( \left(\frac{x^{6}}{9 y^{-12}}\right)^{\frac{1}{2}} \), giving your answer in positive index form. (b) Given that \( x=7^{6 y} \), find an expression in terms of \( x \) for \( 49^{\text {ing } y} \)

To solve the problems, let's break them down step by step. ### Part (a) We need to simplify the expression \( \left(\frac{x^{6}}{9 y^{-12}}\right)^{\frac{1}{2}} \). 1. **Apply the exponent to both the numerator and the denominator:** \[ \left(\frac{x^{6}}{9 y^{-12}}\right)^{\frac{1}{2}} = \frac{(x^{6})^{\frac{1}{2}}}{(9)^{\frac{1}{2}} (y^{-12})^{\frac{1}{2}}} \] 2. **Simplify each part:** - For the numerator: \[ (x^{6})^{\frac{1}{2}} = x^{6 \cdot \frac{1}{2}} = x^{3} \] - For the denominator: \[ (9)^{\frac{1}{2}} = 3 \quad \text{and} \quad (y^{-12})^{\frac{1}{2}} = y^{-12 \cdot \frac{1}{2}} = y^{-6} \] 3. **Combine the results:** \[ \frac{x^{3}}{3 y^{-6}} = \frac{x^{3}}{3} \cdot y^{6} = \frac{x^{3} y^{6}}{3} \] Thus, the simplified expression in positive index form is: \[ \frac{x^{3} y^{6}}{3} \] ### Part (b) We are given that \( x = 7^{6y} \) and need to find an expression for \( 49^{\text{ing } y} \) in terms of \( x \). 1. **Rewrite \( 49 \) in terms of base \( 7 \):** \[ 49 = 7^{2} \] 2. **Express \( 49^{y} \):** \[ 49^{y} = (7^{2})^{y} = 7^{2y} \] 3. **Now, we need to express \( 7^{2y} \) in terms of \( x \):** - Since \( x = 7^{6y} \), we can express \( 7^{2y} \) in terms of \( x \): \[ 7^{2y} = 7^{\frac{2}{6} \cdot 6y} = 7^{\frac{1}{3} \cdot 6y} = (7^{6y})^{\frac{1}{3}} = x^{\frac{1}{3}} \] Thus, the expression for \( 49^{y} \) in terms of \( x \) is: \[ 49^{y} = x^{\frac{1}{3}} \] ### Final Answers (a) \( \frac{x^{3} y^{6}}{3} \) (b) \( 49^{y} = x^{\frac{1}{3}} \)