The given expression is:

$\left( \frac{4x^5y}{16xy^4} \right)^3$

Let’s simplify this step by step.

Step 1: Simplify the fraction inside the parentheses

We have:

$\frac{4x^5y}{16xy^4}$

First, simplify the coefficients:
$\frac{4}{16} = \frac{1}{4}$

Now, simplify the variables:

• For $x^5$ and $x$, apply the law of exponents: $\frac{x^5}{x} = x^{5-1} = x^4$
• For $y$ and $y^4$, apply the law of exponents: $\frac{y}{y^4} = y^{1-4} = y^{-3}$

So the simplified expression becomes: $\frac{1}{4}x^4y^{-3}$

Step 2: Apply the cube (raise to the power of 3)

Now, raise the simplified expression to the power of 3: $\left( \frac{1}{4}x^4y^{-3} \right)^3$

Apply the power of 3 to each part:

• Coefficient: $\left(\frac{1}{4}\right)^3 = \frac{1}{64}$
• $x^4$ raised to the power of 3: $(x^4)^3 = x^{4 \times 3} = x^{12}$
• $y^{-3}$ raised to the power of 3: $(y^{-3})^3 = y^{-9}$

So the final simplified expression is: $\frac{x^{12}}{64y^9}$

This is the fully simplified result.

Would you like any further clarification or details on these steps? Here are some related questions to deepen your understanding:

1. How do you simplify expressions with negative exponents?
2. What is the rule for raising a fraction to a power?
3. How can you generalize the power rule for more complex expressions?
4. What happens if you have multiple variables with different exponents?
5. Can you explain how coefficients are handled differently than variables in exponentiation?

Tip: Always apply exponent rules carefully, starting with simplifying coefficients and then using exponent rules for variables.