Let's solve the given problem step by step. The expression is:

$\left( \frac{\sqrt{64a^{12}}}{(d^{1.5})^{-6}} \right)^{\frac{2}{3}}$

Step 1: Simplify the square root in the numerator

$\sqrt{64a^{12}} = \sqrt{64} \cdot \sqrt{a^{12}} = 8a^6$

So the numerator becomes: $8a^6$

Step 2: Simplify the denominator

The denominator is $(d^{1.5})^{-6}$. Using the property of exponents $(x^m)^n = x^{m \cdot n}$, we get: $(d^{1.5})^{-6} = d^{1.5 \cdot (-6)} = d^{-9}$

So the denominator becomes: $d^{-9}$

Step 3: Combine numerator and denominator

$\frac{\sqrt{64a^{12}}}{(d^{1.5})^{-6}} = \frac{8a^6}{d^{-9}}$ Since $d^{-9}$ in the denominator moves to the numerator as $d^9$: $\frac{8a^6}{d^{-9}} = 8a^6d^9$

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

We now raise $8a^6d^9$ to the power of $\frac{2}{3}$: $(8a^6d^9)^{\frac{2}{3}}$

Using the property of exponents $(x^m y^n z^p)^q = x^{m \cdot q} y^{n \cdot q} z^{p \cdot q}$, we distribute the exponent $\frac{2}{3}$ to each term: $8^{\frac{2}{3}} \cdot a^{6 \cdot \frac{2}{3}} \cdot d^{9 \cdot \frac{2}{3}}$

Step 5: Simplify each term

1. $8^{\frac{2}{3}}$: Since $8 = 2^3$, we have: $8^{\frac{2}{3}} = (2^3)^{\frac{2}{3}} = 2^{3 \cdot \frac{2}{3}} = 2^2 = 4$

2. $a^{6 \cdot \frac{2}{3}} = a^4$

3. $d^{9 \cdot \frac{2}{3}} = d^6$

Final Simplified Expression:

$4a^4d^6$

Let me know if you'd like further clarification or details! Below are related practice questions for additional understanding.

Related Questions:

1. Simplify $(x^4 y^{-2})^{\frac{1}{2}}$.
2. Rewrite $(27a^3b^6)^{\frac{2}{3}}$ in simplified form.
3. Simplify $\frac{\sqrt{16x^{10}}}{y^{-4}}$.
4. Evaluate $(125m^9n^{-12})^{\frac{1}{3}}$.
5. Simplify $(x^{0.5}y^{-2})^{-3}$.

Tip:

When simplifying exponents, always apply the rules of powers systematically (e.g., product, quotient, and power rules) to avoid mistakes.