https://file.aiquickdraw.com/mathbot/file/1729703781308iwz7hd5i.jpg

The expression in the image 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:

\[
\frac{4x^5y}{16xy^4}
\]

Simplifying the numerical coefficient:  
\[
\frac{4}{16} = \frac{1}{4}
\]

Now simplify the powers of \(x\) and \(y\):

- For \(x\), the powers are \(x^5\) in the numerator and \(x\) (which is \(x^1\)) in the denominator. Using the rule of exponents (\(x^a / x^b = x^{a-b}\)):
  \[
  x^5 / x^1 = x^{5-1} = x^4
  \]
- For \(y\), the powers are \(y\) (which is \(y^1\)) in the numerator and \(y^4\) in the denominator. Similarly:
  \[
  y^1 / y^4 = y^{1-4} = y^{-3}
  \]

So the expression simplifies to:
\[
\frac{1}{4}x^4y^{-3}
\]

### Step 2: Apply the cube
Now we need to raise the entire simplified expression to the power of 3:

\[
\left( \frac{1}{4}x^4y^{-3} \right)^3
\]

Distribute the power to each term:
- For \(\frac{1}{4}\), raise to the power of 3:
  \[
  \left(\frac{1}{4}\right)^3 = \frac{1^3}{4^3} = \frac{1}{64}
  \]
- For \(x^4\), raise to the power of 3:
  \[
  (x^4)^3 = x^{4 \cdot 3} = x^{12}
  \]
- For \(y^{-3}\), raise to the power of 3:
  \[
  (y^{-3})^3 = y^{-3 \cdot 3} = y^{-9}
  \]

Thus, the entire expression becomes:

\[
\frac{1}{64}x^{12}y^{-9}
\]

### Final Answer:
\[
\frac{x^{12}}{64y^9}
\]

Let me know if you need further details or clarification.

---

Here are 5 related questions to explore:

1. How do you apply the rules of exponents when dividing terms with the same base?
2. What happens when you raise a fraction to a power?
3. How would the result change if the expression were squared instead of cubed?
4. Can you simplify \( \left( \frac{2x^3}{8xy^2} \right)^4 \)?
5. How does the exponent rule apply when negative exponents are involved?

**Tip:** Always simplify the inner expression first before applying exponents to avoid confusion.

(4x^5 * y) / (16xy^4) raised to the power of 3

This detailed solution walks through simplifying the algebraic fraction (4x^5y)/(16xy^4) and raising it to the power of 3. It covers key exponent rules, including simplifying powers and fractions. Suitable for Grades 8-10 students.

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