https://file.aiquickdraw.com/mathbot/file/1729431831872z0bm0l8q.jpg

The expression in the image is:

\[
\left( \frac{256x^{20}}{y^8} \right)^{-\frac{1}{4}}
\]

To simplify this expression, we can apply the power of a fraction rule, which states that:

\[
\left( \frac{a}{b} \right)^n = \frac{a^n}{b^n}
\]

We can also apply the power of a power rule, which says:

\[
(x^m)^n = x^{m \cdot n}
\]

### Step 1: Apply the exponent to the numerator and denominator.
The expression becomes:

\[
\frac{(256x^{20})^{-\frac{1}{4}}}{(y^8)^{-\frac{1}{4}}}
\]

### Step 2: Simplify the numerator and denominator.
- \(256^{-\frac{1}{4}} = \left( 2^8 \right)^{-\frac{1}{4}} = 2^{-2} = \frac{1}{4}\)
- \(x^{20 \cdot -\frac{1}{4}} = x^{-5}\)
- \(y^{8 \cdot -\frac{1}{4}} = y^{-2}\)

Now the expression becomes:

\[
\frac{\frac{1}{4} x^{-5}}{y^{-2}}
\]

### Step 3: Simplify the division of exponents.
We can simplify this to:

\[
\frac{1}{4} \cdot \frac{x^{-5}}{y^{-2}} = \frac{1}{4} \cdot \frac{y^2}{x^5}
\]

### Final Expression:

\[
\frac{y^2}{4x^5}
\]

This is the simplified form of the given expression.

Would you like more details, or do you have any questions?

Here are some related questions you might find helpful:
1. How do negative exponents work in algebraic expressions?
2. How can you further simplify expressions involving roots and exponents?
3. What is the general rule for distributing exponents over multiplication and division?
4. How do fractional exponents relate to roots?
5. Can expressions like these be generalized to higher powers or more variables?

**Tip:** Always simplify powers of constants first to make the overall simplification process easier.

Learn how to simplify the expression (256x^20 / y^8)^(-1/4) step by step. This problem explores key algebraic concepts, such as negative exponents, fractional exponents, and exponent rules. Suitable for high school students in Grades 9-12.

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