Hand writing a math problem with exponents on paper.

Let’s solve this step by step.

We are given the expression:

\[
\frac{7^{-4} x^{8} y^{-6}}{x^{-5} x^{-10} y^{-20}}
\]

Wait — looking again, in the denominator it says:
“x⁻⁵ x⁻¹⁰ y⁻²⁰” — that means we have two x terms multiplied together in the denominator.

So first, let’s simplify the denominator.

In the denominator:
\( x^{-5} \cdot x^{-10} = x^{-5 + (-10)} = x^{-15} \)

So now the whole expression becomes:

\[
\frac{7^{-4} x^{8} y^{-6}}{x^{-15} y^{-20}}
\]

Now, we can separate the constants and variables:

= \( 7^{-4} \cdot \frac{x^{8}}{x^{-15}} \cdot \frac{y^{-6}}{y^{-20}} \)

Now use the rule: \( \frac{a^m}{a^n} = a^{m - n} \)

For x:
\( x^{8 - (-15)} = x^{8 + 15} = x^{23} \)

For y:
\( y^{-6 - (-20)} = y^{-6 + 20} = y^{14} \)

And for the constant:
\( 7^{-4} = \frac{1}{7^4} \)

But wait — is there any other number? The original numerator has only 7⁻⁴, no other numbers. So we leave it as is unless told to write without negative exponents.

The problem doesn’t specify, but usually we want positive exponents in final answer.

So:

Final expression:

\[
\frac{x^{23} y^{14}}{7^4}
\]

Now calculate 7⁴:

7² = 49
7⁴ = (7²)² = 49² = 2401

So:

\[
\frac{x^{23} y^{14}}{2401}
\]

✔ Let’s double-check:

Original:
Numerator: 7⁻⁴ x⁸ y⁻⁶
Denominator: x⁻⁵ * x⁻¹⁰ * y⁻²⁰ → x⁻¹⁵ y⁻²⁰

So dividing:

7⁻⁴ stays
x⁸ / x⁻¹⁵ = x²³
y⁻⁶ / y⁻²⁰ = y¹⁴

Yes.

And 7⁻⁴ = 1/2401

So final answer is:

\[
\frac{x^{23} y^{14}}{2401}
\]

Final Answer:
\[
\frac{x^{23} y^{14}}{2401}
\]