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To solve for \( x \) in the equation:

\[
\left( \frac{1}{4} \right)^{x - 2} = 16
\]

we can proceed with these steps:

1. Rewrite \( \frac{1}{4} \) and \( 16 \) as powers of \( 2 \):
   - \( \frac{1}{4} = 2^{-2} \)
   - \( 16 = 2^4 \)

   Substituting, we get:
   \[
   (2^{-2})^{x - 2} = 2^4
   \]

2. Apply the power rule \((a^m)^n = a^{m \cdot n}\):
   \[
   2^{-2(x - 2)} = 2^4
   \]

3. Since the bases are the same, set the exponents equal to each other:
   \[
   -2(x - 2) = 4
   \]

4. Distribute the \(-2\):
   \[
   -2x + 4 = 4
   \]

5. Subtract \(4\) from both sides:
   \[
   -2x = 0
   \]

6. Divide by \(-2\):
   \[
   x = 0
   \]

So, the solution is:
\[
x = 0
\]

Would you like more details on any of the steps? 

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### Related Questions:
1. How would the solution change if the exponent were \(x + 2\) instead of \(x - 2\)?
2. Can we solve similar problems using logarithms?
3. What if the base was different, such as \(\frac{1}{3}\)?
4. How can we verify that \(x = 0\) is indeed the correct solution?
5. How would we solve if we had \(\left(\frac{1}{4}\right)^{x-2} = \frac{1}{16}\) instead of \(16\)?

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**Tip:** Always try to express numbers with the same base to simplify exponential equations.

Solve for x.

\( \left( \frac{1}{4} \right)^{x - 2} = 16 \)

Discover how to solve the exponential equation \( \left( \frac{1}{4} \right)^{x - 2} = 16 \) with a clear, step-by-step solution. This problem demonstrates key algebraic concepts such as exponentiation and properties of exponents, ideal for students in Grades 9-11.

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