The expression provided is:

$\left(\frac{x^4}{2^{10}}\right)^{\frac{1}{2}}$

Let's simplify this step by step:

1. Apply the exponent to both the numerator and denominator:
   Since the expression inside the parentheses is raised to the power of $\frac{1}{2}$, we can distribute this exponent to both the numerator and the denominator:

   $\left(\frac{x^4}{2^{10}}\right)^{\frac{1}{2}} = \frac{\left(x^4\right)^{\frac{1}{2}}}{\left(2^{10}\right)^{\frac{1}{2}}}$

2. Simplify each part:

   • For the numerator $\left(x^4\right)^{\frac{1}{2}}$, we use the rule $(a^m)^n = a^{mn}$:

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

   • For the denominator $\left(2^{10}\right)^{\frac{1}{2}}$, apply the same rule:

     $\left(2^{10}\right)^{\frac{1}{2}} = 2^{10 \times \frac{1}{2}} = 2^5 = 32$

3. Final simplified expression:

   $\frac{x^2}{32}$

So, the simplified expression is:

$\boxed{\frac{x^2}{32}}$

Would you like more details or have any questions?

Here are some related questions you might find helpful:

1. What happens if the exponent was a different fraction, like $\frac{3}{2}$?
2. How do you simplify expressions with variables in both the numerator and denominator?
3. What are the general rules for simplifying exponents?
4. How would you simplify the expression if it were inside a square root?
5. What if the base of the denominator was a variable instead of a number?

Tip: Always remember to distribute the exponent to both the numerator and the denominator when simplifying expressions with fractions raised to a power.