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} =