https://r2.erweima.ai/mathbot/image/1727022195816ocm2zxnm.jpg

The expression given in the image is:

\[
\left( \frac{\frac{1}{4} \cdot \sqrt{6} \cdot 2 \cdot \sqrt{3} \cdot \sqrt{2}}{\sqrt{27}} \right)^{-2}
\]

Let's break it down and simplify the steps:

### Step 1: Simplify the numerator
The numerator is:
\[
\frac{1}{4} \cdot \sqrt{6} \cdot 2 \cdot \sqrt{3} \cdot \sqrt{2}
\]

- Combine the constants: \(\frac{1}{4} \cdot 2 = \frac{1}{2}\)
- Combine the square roots: \(\sqrt{6} \cdot \sqrt{3} \cdot \sqrt{2}\)

Using the property \(\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}\), we get:
\[
\sqrt{6} \cdot \sqrt{3} \cdot \sqrt{2} = \sqrt{6 \cdot 3 \cdot 2} = \sqrt{36} = 6
\]

Thus, the numerator becomes:
\[
\frac{1}{2} \cdot 6 = 3
\]

### Step 2: Simplify the denominator
The denominator is \(\sqrt{27}\), and we know:
\[
\sqrt{27} = \sqrt{9 \cdot 3} = \sqrt{9} \cdot \sqrt{3} = 3\sqrt{3}
\]

### Step 3: Simplify the entire fraction
The entire fraction is now:
\[
\frac{3}{3\sqrt{3}} = \frac{1}{\sqrt{3}}
\]

### Step 4: Apply the exponent
We now have the expression:
\[
\left( \frac{1}{\sqrt{3}} \right)^{-2}
\]

Using the property \(\left( \frac{1}{x} \right)^{-n} = x^n\), we get:
\[
\left( \frac{1}{\sqrt{3}} \right)^{-2} = \left( \sqrt{3} \right)^2 = 3
\]

### Final Answer:
The value of the expression is:
\[
\boxed{3}
\]

Would you like more details or have any questions?

Here are 5 related questions to explore:

1. What are the properties of exponents and radicals used in this solution?
2. How do you simplify expressions with nested square roots?
3. What happens when negative exponents are applied to fractions?
4. Can this expression be further simplified if it was raised to another power?
5. How can we verify this result using an alternative method?

**Tip:** Always simplify the radicals and fractions before applying exponents, as this makes the calculations easier to manage.

Calculate the expression: ( (1/4) * sqrt(6) * 2 * sqrt(3) * sqrt(2) / sqrt(27) )^(-2)

Learn how to simplify the expression ( (1/4) * sqrt(6) * 2 * sqrt(3) * sqrt(2) / sqrt(27) )^(-2) step-by-step. This problem covers the laws of exponents and properties of radicals, suitable for high school students in Grades 9-12.

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