Let's solve each question step-by-step.

Problem 9

Given: $\sqrt{\frac{a^4}{5} \cdot \frac{b^{-2}}{3}} = a^x b^y$

Step 1: Simplify the expression inside the square root

Rewrite the expression under the square root: $\sqrt{\frac{a^4}{5} \cdot \frac{b^{-2}}{3}} = \sqrt{\frac{a^4 b^{-2}}{15}}$

Step 2: Apply the square root to each part

Using $\sqrt{\frac{a^4 b^{-2}}{15}} = \frac{a^2 b^{-1}}{\sqrt{15}}$, we get: $\frac{a^2 b^{-1}}{\sqrt{15}} = a^x b^y$

Step 3: Equate powers of $a$ and $b$

Since both sides must be equal, we can compare the exponents of $a$ and $b$:

1. For $a$: $x = 2$
2. For $b$: $y = -1$

Thus, $x = 2$ and $y = -1$.

Problem 10

Given: $\left(\frac{a^x}{b^{5-x}}\right) \cdot \frac{b^{y-4}}{a^y} = a^2 b^4$

Step 1: Combine the terms

Combine the terms by bringing $a$ and $b$ terms together: $\frac{a^x}{b^{5-x}} \cdot \frac{b^{y-4}}{a^y} = \frac{a^{x-y} b^{y-4}}{b^{5-x}}$

Step