https://file.aiquickdraw.com/mathbot/file/17265432491315xujwdrw.jpg

Let's work through this step by step.

The expression given is:

\[
\left( \frac{5x^3}{2y^8} \right)^4 \cdot \frac{2^2 y^6}{5^3 x^7}
\]

### Step 1: Simplify the first fraction raised to the 4th power.

We apply the exponent of 4 to each component inside the parentheses:

\[
\left( \frac{5x^3}{2y^8} \right)^4 = \frac{5^4 x^{3 \cdot 4}}{2^4 y^{8 \cdot 4}} = \frac{625 x^{12}}{16 y^{32}}
\]

### Step 2: Multiply with the second fraction.

Now, we multiply:

\[
\frac{625 x^{12}}{16 y^{32}} \cdot \frac{2^2 y^6}{5^3 x^7}
\]

This becomes:

\[
\frac{625 x^{12} \cdot 2^2 y^6}{16 y^{32} \cdot 5^3 x^7}
\]

### Step 3: Simplify the expression.

- \( 625 = 5^4 \), and \( 5^4 \cdot 5^{-3} = 5^{4-3} = 5^1 \)
- \( 2^2 = 4 \), and \( 4 \cdot 625 = 2500 \)
- Combine the powers of \( x \): \( x^{12} \cdot x^{-7} = x^{12-7} = x^5 \)
- Combine the powers of \( y \): \( y^6 \cdot y^{-32} = y^{6-32} = y^{-26} \)

Thus, the result becomes:

\[
\frac{2500 x^5}{16 y^{26}}
\]

### Final Answer:

- **Numerator**: \( 2500 x^5 \)
- **Denominator**: \( 16 y^{26} \)

Would you like a more detailed explanation or help with any other steps?

Here are 5 related questions for practice:
1. How do you simplify expressions with negative exponents?
2. What are the laws of exponents and how are they applied in simplification?
3. How do you simplify a fraction raised to a power?
4. How do you multiply two fractions with exponents?
5. What is the process for eliminating negative exponents?

**Tip:** Always apply the exponent rules carefully when simplifying terms to avoid mistakes with negative or fractional exponents.

Multiply and simplify completely, including the elimination of all negative exponents: (5x^3 / 2y^8)^4 * (2^2 y^6 / 5^3 x^7)

Learn how to multiply and simplify the exponential expression (5x^3 / 2y^8)^4 * (2^2 y^6 / 5^3 x^7) with detailed steps. This problem involves exponentiation rules and fraction multiplication, suitable for Grades 9-11 students.

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