The expression given in the image is:

$(64x^4)^{0.5} \times 4x^{-2}$

Let's solve it step by step.

Step 1: Simplify $(64x^4)^{0.5}$

We can apply the exponent rule $(a^m \cdot b^m) = (ab)^m$:

$(64x^4)^{0.5} = 64^{0.5} \cdot (x^4)^{0.5}$

Now, simplifying both parts: $64^{0.5} = \sqrt{64} = 8$ $(x^4)^{0.5} = x^{4 \cdot 0.5} = x^2$

Thus, we get: $(64x^4)^{0.5} = 8x^2$

Step 2: Multiply by $4x^{-2}$

Now, multiply the result by $4x^{-2}$:

$8x^2 \times 4x^{-2} = (8 \times 4) \cdot (x^2 \cdot x^{-2}) = 32 \cdot x^{2 + (-2)} = 32 \cdot x^0$

Since $x^0 = 1$, this simplifies to: $32 \cdot 1 = 32$

Final Answer:

The value of the expression is $32$.

Would you like more details or have any questions? Here are 5 related questions you might consider:

1. What are the general rules for simplifying expressions with exponents?
2. How does the exponent rule $(ab)^m = a^m \cdot b^m$ work in other cases?
3. What is the significance of $x^0 = 1$ for all non-zero $x$?
4. How do fractional exponents relate to roots (e.g., $a^{0.5} = \sqrt{a}$)?
5. Can you explain how negative exponents affect multiplication of terms?

Tip: Remember that multiplying powers of the same base adds their exponents!