https://file.aiquickdraw.com/mathbot/file/17302274369761iewdbgm.jpg

The equation in the image is:

\[
(16^2)^{-\frac{1}{4}} = 64^{\frac{1}{4}} + 4^{x+2}
\]

To solve for \( x \), let's break down each component.

1. **Simplify the Left Side:**  
   \((16^2)^{-\frac{1}{4}}\)
   - Rewrite \( 16 \) as \( 2^4 \):  
     \[
     (16^2)^{-\frac{1}{4}} = ((2^4)^2)^{-\frac{1}{4}} = (2^8)^{-\frac{1}{4}}
     \]
   - Applying the exponent rule \((a^m)^n = a^{m \cdot n}\):
     \[
     (2^8)^{-\frac{1}{4}} = 2^{8 \cdot -\frac{1}{4}} = 2^{-2} = \frac{1}{4}
     \]

   So, the left side simplifies to \( \frac{1}{4} \).

2. **Simplify the Right Side:**  
   \[
   64^{\frac{1}{4}} + 4^{x+2}
   \]
   - Rewrite \( 64 \) as \( 2^6 \):  
     \[
     64^{\frac{1}{4}} = (2^6)^{\frac{1}{4}} = 2^{6 \cdot \frac{1}{4}} = 2^{\frac{3}{2}} = \sqrt{8} = 

Explore a detailed solution to the exponential equation (16^2)^(-1/4) = 64^(1/4) + 4^(x+2). This problem involves the simplification of powers and exponent rules, suitable for high school students in Grades 9-12.

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