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Let's break down the questions from your image and solve them one by one.

### Q1: Solve for \( a \)
You are given:
\[
32 \sqrt{2} = 2^a
\]

First, let's express 32 and \(\sqrt{2}\) as powers of 2:
\[
32 = 2^5
\quad \text{and} \quad \sqrt{2} = 2^{1/2}
\]
So,
\[
32 \sqrt{2} = 2^5 \times 2^{1/2} = 2^{5 + 1/2} = 2^{5.5} = 2^{11/2}
\]

Therefore, equating the powers of 2, we have:
\[
2^a = 2^{11/2}
\]
Thus, \( a = \frac{11}{2} \).

### Q2(a): Evaluate \( 32^{\frac{3}{5}} \)
We already know that \( 32 = 2^5 \). So,
\[
32^{\frac{3}{5}} = (2^5)^{\frac{3}{5}} = 2^{5 \times \frac{3}{5}} = 2^3 = 8
\]

### Q2(b): Simplify fully \( \left( \frac{25x^4}{4} \right)^{\frac{1}{2}} \)
We can apply the square root to both the numerator and the denominator:
\[
\left( \frac{25x^4}{4} \right)^{\frac{1}{2}} = \frac{\sqrt{25x^4}}{\sqrt{4}}
\]
\[
= \frac{5x^2}{2}
\]

Thus, the simplified form is:
\[
\frac{5x^2}{2}
\]

Would you like further clarification or more details on any of the solutions? Here are some related questions you might find useful:

1. How do you convert radical expressions into exponents?
2. Why does \( 32 \) equal \( 2^5 \)?
3. What happens when you multiply exponents with the same base?
4. Can you generalize the simplification process for other fractional powers?
5. How would you handle expressions where both the base and the exponent are variables?

**Tip:** When dealing with fractional exponents, remember that the numerator represents the power, and the denominator represents the root!

Given that 32√2 = 2^a, find the value of a. Also, evaluate 32^(3/5) and simplify (25x^4 / 4)^(1/2).

Solve exponent equations like 32√2 = 2^a and evaluate expressions with fractional exponents, such as 32^(3/5). This problem explains key algebraic concepts and provides step-by-step solutions. Suitable for students in Grades 9-11.

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