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Let's break down the expression step by step and solve it:

The given expression is:

\[
\sqrt{ \left( 216^{\frac{2}{3}} \times 125^{\frac{1}{2}} \right) \div \left( 0.04^{-\frac{3}{2}} \right) }
\]

### Step 1: Evaluate \( 216^{\frac{2}{3}} \)
- \( 216 = 6^3 \), so \( 216^{\frac{1}{3}} = 6 \).
- Therefore, \( 216^{\frac{2}{3}} = 6^2 = 36 \).

### Step 2: Evaluate \( 125^{\frac{1}{2}} \)
- \( 125 = 5^3 \), so \( 125^{\frac{1}{2}} = \sqrt{125} \approx 11.18 \).

### Step 3: Evaluate \( 0.04^{-\frac{3}{2}} \)
- \( 0.04 = \left( \frac{1}{25} \right) \), and \( 25 = 5^2 \), so \( 0.04^{\frac{1}{2}} = \frac{1}{5} \).
- Therefore, \( 0.04^{-\frac{3}{2}} = \left( \frac{1}{5} \right)^{-3} = 5^3 = 125 \).

### Step 4: Combine the values
Now substitute these evaluated terms into the original expression:
\[
\sqrt{ \left( 36 \times 11.18 \right) \div 125 }
\]

### Step 5: Simplify
First, multiply \( 36 \times 11.18 \approx 402.48 \). Now divide by 125:
\[
\frac{402.48}{125} \approx 3.22
\]

### Step 6: Final Step
Now take the square root:
\[
\sqrt{3.22} \approx 1.79
\]

Thus, the final result is approximately **1.79**.

Would you like more details or explanations on any part of the solution?

### Relative Questions:
1. How do you simplify expressions with fractional exponents?
2. What is the general process for handling negative exponents?
3. Can you apply properties of square roots and cube roots together in an expression?
4. What happens when you divide by a negative exponent?
5. How do radical expressions relate to fractional exponents?

### Tip:
When dealing with fractional exponents, always try to simplify using properties of exponents before performing the final computation.

Solve the expression: sqrt((216^(2/3) * 125^(1/2)) / (0.04^(-3/2)))

Learn how to solve an advanced expression involving fractional exponents and radicals: sqrt((216^(2/3) * 125^(1/2)) / (0.04^(-3/2))). The solution includes key exponent rules and radical simplification techniques. Suitable for Grades 9-12.

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