Let’s work through this problem to ensure the solution is correct and matches your source.

The given expression is:
$\left( \frac{\sqrt{64a^{12}}}{(a^{1.5})^{-6}} \right)^{\frac{2}{3}}$

Step-by-step solution:

1. Simplify the square root in the numerator:
   $\sqrt{64a^{12}} = \sqrt{64} \cdot \sqrt{a^{12}} = 8a^6$

   So the expression becomes: $\left( \frac{8a^6}{(a^{1.5})^{-6}} \right)^{\frac{2}{3}}$

2. Simplify the denominator:
   $(a^{1.5})^{-6} = a^{1.5 \cdot -6} = a^{-9}$

   This makes the fraction: $\frac{8a^6}{a^{-9}} = 8a^{6 - (-9)} = 8a^{15}$

3. Raise the result to the $\frac{2}{3}$ power**:
   $\left(8a^{15}\right)^{\frac{2}{3}} = 8^{\frac{2}{3}} \cdot \left(a^{15}\right)^{\frac{2}{3}}$

   • $8^{\frac{2}{3}} = (\sqrt[3]{8})^2 = 2^2 = 4$
   • $\left(a^{15}\right)^{\frac{2}{3}} = a^{15 \cdot \frac{2}{3}} = a^{10}$

   Combining these results: $4a^{10}$

Final Answer:

$4a^{10}$

Confirmation:

This matches the answer provided by your source. The variable in the denominator was corrected to $a$, and the calculations confirm the result.

Would you like any clarification or a breakdown of any specific step?

Related Questions:

1. What is the general rule for simplifying fractional exponents like $(x^a)^b$?
2. How is the cube root of a number simplified, as in $8^{2/3}$?
3. What happens when a base with an exponent is raised to a negative power?
4. Can we apply the $\sqrt{}$ rule directly if the exponent isn’t even, such as $a^{11.5}$?
5. How does the distributive property of exponents work in terms of $a^m \cdot a^n$?

Tip:

Always check for any errors in interpreting negative exponents or fractional powers—they often simplify the most!