https://r2.erweima.ai/mathbot/image/1726166698837kxqvhptn.jpg

The expression shown in the image is:

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

Let’s break this down step by step:

1. The innermost part of the expression is \(-\frac{1}{4}\), raised to the power of \(-4\). This can be written as:
   \[
   \left( -\frac{1}{4} \right)^{-4}
   \]

   When a fraction is raised to a negative exponent, we first take the reciprocal and then raise it to the positive exponent. So:
   \[
   \left( -\frac{1}{4} \right)^{-4} = \left( -4 \right)^4 = 4^4 = 256
   \]
   (since \((-4)^4\) equals \(256\)).

2. Now we move on to the next part of the expression, where we have:
   \[
   -\left( 256 \right)^3
   \]
   Cubing 256 gives:
   \[
   256^3 = 16777216
   \]
   So this becomes:
   \[
   -16777216
   \]

3. Finally, the entire expression is raised to the power of \(-1\), which means we take the reciprocal of \(-16777216\):
   \[
   \left( -16777216 \right)^{-1} = -\frac{1}{16777216}
   \]

Thus, the value of the entire expression simplifies to:

\[
-\frac{1}{16777216}
\]

Would you like further details or any specific questions on this? Here are some related questions:

1. What happens when you raise a negative fraction to an even exponent?
2. How do negative exponents affect the terms inside a fraction?
3. Why is the reciprocal taken when raising something to the power of \(-1\)?
4. How does cubing a number like \(256\) change the result?
5. Can this process be applied to more complex fractions or different numbers?

**Tip:** Always simplify expressions by working from the innermost parentheses outward and handling exponents carefully, especially negative ones!

Simplify the mathematical expression involving nested fractions and exponents, as seen in the image.

Learn how to simplify expressions with nested fractions and negative exponents. This problem covers exponentiation, negative exponents, and handling complex algebraic expressions. Suitable for students in Grades 10-12.

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