https://file.aiquickdraw.com/mathbot/file/1725426450933vi8war15.jpg

The expression in the image is:

\[
\left(\frac{1}{4x^8}\right)^{-3}
\]

To simplify this expression, we can use the property of exponents that states:

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

Applying this property to your expression:

\[
\left(\frac{1}{4x^8}\right)^{-3} = \left(\frac{4x^8}{1}\right)^{3} = (4x^8)^3
\]

Now, simplify further by applying the power to both the constant and the variable:

\[
(4x^8)^3 = 4^3 \cdot (x^8)^3 = 64 \cdot x^{24}
\]

So the simplified expression is:

\[
64x^{24}
\]

Would you like further details or have any questions about this solution? Here are five related questions for further exploration:

1. How do you simplify expressions with negative exponents?
2. What happens when you raise a power to another power?
3. How do you distribute an exponent over a product?
4. What is the result of simplifying \(\left(\frac{1}{2x^5}\right)^{-2}\)?
5. How can you rewrite \(\frac{1}{x^{-2}}\) using positive exponents?

**Tip:** Remember that when dealing with negative exponents, inverting the base (flipping the fraction) and making the exponent positive simplifies the expression effectively.

Learn how to simplify the expression (1 / 4x^8) ^ -3 using algebraic principles and exponentiation rules. This problem involves understanding negative exponents, the power of a fraction, and how to simplify complex expressions. Suitable for students in Grades 8-10.

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