The given expression is:
$\left( \frac{\sqrt{64a^{12}}}{(d^{1.5})^{-6}} \right)^{\frac{2}{3}}$
We will simplify this step by step.

Step 1: Simplify the square root $\sqrt{64a^{12}}$

The square root of $64a^{12}$ can be written as:
$\sqrt{64a^{12}} = \sqrt{64} \cdot \sqrt{a^{12}} = 8a^6.$

Step 2: Simplify the denominator $(d^{1.5})^{-6}$

Using the rule $(x^a)^b = x^{a \cdot b}$, we have:
$(d^{1.5})^{-6} = d^{1.5 \cdot (-6)} = d^{-9}.$
This simplifies the denominator to:
$d^{-9}.$

Step 3: Combine numerator and denominator

After simplifying the square root and the denominator, the expression becomes:
$\frac{8a^6}{d^{-9}}.$
Since $d^{-9}$ in the denominator becomes $d^9$ in the numerator, this simplifies to:
$8a^6d^9.$

Step 4: Apply the outer exponent $\frac{2}{3}$

Raise the entire expression $8a^6d^9$ to the power of $\frac{2}{3}$:
$(8a^6d^9)^{\frac{2}{3}}.$

Using the rule $(x \cdot y \cdot z)^a = x^a \cdot y^a \cdot z^a$, we apply $\frac{2}{3}$ to each factor:
$(8)^{\frac{2}{3}} \cdot (a^6)^{\frac{2}{3}} \cdot (d^9)^{\frac{2}{3}}.$

Step 5: Simplify each term

1. $8^{\frac{2}{3}} = (2^3)^{\frac{2}{3}} = 2^{2} = 4.$
2. $(a^6)^{\frac{2}{3}} = a^{6 \cdot \frac{2}{3}} = a^4.$
3. $(d^9)^{\frac{2}{3}} = d^{9 \cdot \frac{2}{3}} = d^6.$

Final Answer:

$4a^4d^6.$

Would you like a detailed explanation of any step or further practice with similar problems?

Related Questions:

1. How do fractional exponents work with bases involving square roots?
2. What is the general rule for simplifying $(x^a)^b$?
3. How can cube roots like $x^{1/3}$ be simplified in expressions like this one?
4. What is the significance of positive exponents in expressing the final answer?
5. How can this be extended to expressions with more complex fractional exponents?

Tip:

Always simplify step-by-step, starting with parentheses and roots, before applying outer exponents.