https://file.aiquickdraw.com/mathbot/file/17267934093435ge16j17.jpg

The expression in the image is:

\[
\left( \frac{\frac{1}{P^2 q^{4/3} r}}{\frac{1}{4 p q^4}} \right)^8
\]

We can simplify this step by step.

### Step 1: Simplify the inner fraction

The inner part of the expression is:

\[
\frac{\frac{1}{P^2 q^{4/3} r}}{\frac{1}{4 p q^4}} 
= \frac{1}{P^2 q^{4/3} r} \times \frac{4 p q^4}{1}
\]

This simplifies to:

\[
\frac{4 p q^4}{P^2 q^{4/3} r} 
\]

### Step 2: Simplify powers of \( q \)

Using the property of exponents \( q^a / q^b = q^{a-b} \), we simplify the powers of \( q \):

\[
q^4 / q^{4/3} = q^{4 - 4/3} = q^{12/3 - 4/3} = q^{8/3}
\]

Thus, the expression becomes:

\[
\frac{4 p q^{8/3}}{P^2 r}
\]

### Step 3: Raise the result to the power of 8

Now, we need to raise the entire simplified expression to the power of 8:

\[
\left( \frac{4 p q^{8/3}}{P^2 r} \right)^8 = \frac{(4 p)^8 (q^{8/3})^8}{(P^2)^8 r^8}
\]

### Step 4: Simplify each term

\[
(4 p)^8 = 4^8 p^8
\]
\[
(q^{8/3})^8 = q^{(8/3) \times 8} = q^{64/3}
\]
\[
(P^2)^8 = P^{16}
\]
\[
r^8 = r^8
\]

So, the final expression is:

\[
\frac{4^8 p^8 q^{64/3}}{P^{16} r^8}
\]

### Conclusion:

The result of the given expression is:

\[
\frac{4^8 p^8 q^{64/3}}{P^{16} r^8}
\]

Would you like further details or any questions on this?

Here are some related questions to practice:

1. What are the general properties of exponents in fraction operations?
2. How would you simplify \( (x^m / y^n)^k \) for general values?
3. Can you simplify \( \frac{x}{y} \times \frac{a}{b} \) when both numerator and denominator have common terms?
4. What is the importance of understanding fractional powers like \( q^{4/3} \)?
5. How would the solution change if the exponent outside the parentheses was different, e.g., 5 instead of 8?

**Tip:** When simplifying expressions involving exponents, always combine like terms and manage the powers using exponent rules before expanding.

Berapakah hasil dari \left( \frac{\frac{1}{P^2 q^{4/3} r}}{\frac{1}{4 p q^4}} \right)^8?

Learn how to simplify the algebraic fraction expression \left( \frac{\frac{1}{P^2 q^{4/3} r}}{\frac{1}{4 p q^4}} \right)^8. This problem covers exponents, fractional powers, and simplification rules, making it suitable for high school students in Grades 9-12.

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