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I see that you've uploaded an image containing multiple mathematical expressions to simplify. I will begin simplifying a few of them one by one and provide explanations for each.

Let's start with the first few:

### a. \(\left( x^2 y^2 \right)^{-1} \)

#### Simplification:
\[
\left( x^2 y^2 \right)^{-1} = \frac{1}{x^2 y^2}
\]

### b. \(\frac{x^{-4} y^{-1/5}}{x^2 y^{-3}} \)

#### Simplification:
1. Use the property of exponents: \( \frac{x^a}{x^b} = x^{a-b} \) and \( \frac{y^a}{y^b} = y^{a-b} \)
\[
= x^{-4 - 2} y^{-\frac{1}{5} - (-3)} = x^{-6} y^{-\frac{1}{5} + 3}
\]
\[
= x^{-6} y^{\frac{14}{5}}
\]
This can be written as:
\[
= \frac{y^{\frac{14}{5}}}{x^6}
\]

### c. \(\frac{\left(2x^3\right)^2}{x}\)

#### Simplification:
First, expand the numerator:
\[
(2x^3)^2 = 4x^6
\]
Now divide by \(x\):
\[
\frac{4x^6}{x} = 4x^5
\]

### d. \((4x^3)^2 x\)

#### Simplification:
First, expand:
\[
(4x^3)^2 = 16x^6
\]
Multiply by \(x\):
\[
16x^6 \cdot x = 16x^7
\]

### e. \((-4x)^4 (3x^2)^{-3}\)

#### Simplification:
1. Compute \((-4x)^4\):
\[
(-4x)^4 = 256x^4
\]
2. Compute \((3x^2)^{-3}\):
\[
(3x^2)^{-3} = \frac{1}{27x^6}
\]
Now, multiply the two terms:
\[
256x^4 \cdot \frac{1}{27x^6} = \frac{256}{27} x^{-2}
\]
This can be written as:
\[
\frac{256}{27x^2}
\]

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Would you like me to continue with the next problems or go into more details for any of the steps above? Here are some questions to extend the concepts:

1. How can exponent rules be applied to rational exponents?
2. How do negative exponents behave in terms of reciprocals?
3. What happens when you raise a product inside parentheses to a power?
4. How does simplifying expressions with multiple variables and exponents differ from single-variable cases?
5. How do coefficients affect the application of exponent rules?

**Tip:** Always combine like terms before applying exponent rules, as this reduces the complexity of the expression early on.

Simplify the following mathematical expressions, including expressions with exponents, square roots, and variable terms.

This page provides a step-by-step guide to simplifying algebraic expressions involving exponents and radicals. Explore examples like (x^2 y^2)^-1, (4x^3)^2 x, and √(9x^4 y^9), suitable for students in grades 9-12.

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