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

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### 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 \(y = -1\)**.

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### 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 

Given the following math problems:

1. Given that \( \sqrt{\frac{a^4}{5} \cdot \frac{b^{-2}}{3}} = a^x b^y \), find the value of \( x \) and \( y \).
2. Given that \( \left( \frac{a^x}{b^{5-x}} \right) \cdot \frac{b^{y-4}}{a^y} = a^2 b^4 \), find the value of \( x \) and \( y \).
3. Simplify \( (1 + x)^{\frac{3}{2}} - (1 + x)^{\frac{1}{2}} \).

Learn how to solve exponential equations and simplify expressions step-by-step. This problem set covers key algebraic concepts, including the properties of exponents, roots, and simplification techniques. Suitable for students in Grades 10-12.

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