The equation shown in the image is:

$\left(\frac{1}{2}\right)^x = \left(2^{\frac{4}{1}}\right)^{\frac{1}{8}}$

Let's solve this step by step:

1. Simplify the right-hand side:

   • The expression $2^{\frac{4}{1}}$ is simply $2^4$, which equals 16.
   • Now, we have $\left( 2^4 \right)^{\frac{1}{8}} = 16^{\frac{1}{8}}$.

2. Rewrite $16$ as a power of 2:

   • $16 = 2^4$, so we now have $\left( 2^4 \right)^{\frac{1}{8}} = 2^{\frac{4}{8}} = 2^{\frac{1}{2}}$.

3. Now, the equation becomes: $\left(\frac{1}{2}\right)^x = 2^{\frac{1}{2}}$

4. Rewrite the left-hand side:

   • $\frac{1}{2} = 2^{-1}$, so $\left( \frac{1}{2} \right)^x = (2^{-1})^x = 2^{-x}$.

5. Now, we have the equation: $2^{-x} = 2^{\frac{1}{2}}$

6. Since the bases are the same, we can equate the exponents: $-x = \frac{1}{2}$

7. Solving for $x$: $x = -\frac{1}{2}$

Thus, the solution to the equation is:

$x = -\frac{1}{2}$

Would you like a further explanation or have any other questions?

Here are 5 related questions to extend your understanding:

1. What are the properties of exponents that we used to simplify the equation?
2. How would the solution change if the exponent $\frac{1}{8}$ on the right side were different?
3. Can you think of a real-world application where solving exponential equations is useful?
4. What would happen if the base on both sides of the equation were not the same?
5. How would you solve this equation graphically?

Tip: Always check if both sides of an equation can be rewritten with the same base when dealing with exponents; this simplifies solving the equation immensely.