https://file.aiquickdraw.com/mathbot/file/1725880020573d4ppiaxr.jpg

The equation presented in the image is:

\[
\left( \frac{16w^8}{y^{20}} \right)^{-\frac{3}{4}} = \frac{y^m}{nw^p}
\]

We are tasked with simplifying the expression completely. Let's work through the steps:

### Step 1: Apply the Power Rule
When raising a fraction to a power, apply the exponent to both the numerator and denominator:

\[
\left( \frac{16w^8}{y^{20}} \right)^{-\frac{3}{4}} = \frac{(16w^8)^{-\frac{3}{4}}}{(y^{20})^{-\frac{3}{4}}}
\]

### Step 2: Simplify Each Part
Now, simplify the numerator and the denominator separately.

- For the numerator \((16w^8)^{-\frac{3}{4}}\):
  - \(16^{-\frac{3}{4}}\): This is equivalent to \( \left(2^4\right)^{-\frac{3}{4}} = 2^{-3} = \frac{1}{8} \).
  - \(w^8\) raised to the power of \(-\frac{3}{4}\): Apply the power of a power rule: \(w^{8 \cdot -\frac{3}{4}} = w^{-6}\).

Thus, the numerator becomes:
\[
\frac{1}{8} w^{-6}
\]

- For the denominator \((y^{20})^{-\frac{3}{4}}\):
  - Apply the power of a power rule: \(y^{20 \cdot -\frac{3}{4}} = y^{-15}\).

Thus, the denominator becomes:
\[
y^{-15}
\]

### Step 3: Combine Results
Now, combine the simplified numerator and denominator:

\[
\frac{\frac{1}{8} w^{-6}}{y^{-15}} = \frac{y^{15}}{8w^6}
\]

### Step 4: Compare with the Given Form
We are asked to express the result in the form:

\[
\frac{y^m}{nw^p}
\]

Comparing this with our result:

\[
\frac{y^{15}}{8w^6}
\]

We can conclude:
- \(m = 15\)
- \(n = 8\)
- \(p = 6\)

Thus, the simplified expression is:

\[
\frac{y^{15}}{8w^6}
\]

### Final Answer
The values are:
\[
m = 15, \quad n = 8, \quad p = 6
\]

---

Would you like further details or have any questions? Here are some related questions:

1. What is the power of a quotient rule, and how does it apply in expressions like these?
2. How can you use fractional exponents to represent roots and powers at the same time?
3. Why is the base \(16\) written as \(2^4\), and how does that help with simplifying exponents?
4. How do negative exponents affect the placement of variables in fractions?
5. Can you provide an example where a similar expression has a positive fractional exponent?

**Tip:** Always apply the exponent rules carefully, especially when dealing with fractions and negative exponents, to avoid mistakes in simplification.

Simplify completely: (16w^8 / y^20)^(-3/4) = y^m / nw^p

Learn how to simplify the expression (16w^8 / y^20)^(-3/4) by applying the power of a quotient rule, negative exponents, and power rules. This problem covers key algebraic concepts and is suitable for students in Grades 9-11.

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