The value of `[((sqrt6)^5xx(sqrt6)^(-3))/((sqrt6)^(-2))]^(3//2)` is

Options

A. 216
B. 36
C. `1/36`
D. 1296

Answer

To solve the expression \[ \left[ \frac{(\sqrt{6})^5 \times (\sqrt{6})^{-3}}{(\sqrt{6})^{-2}} \right]^{\frac{3}{2}} \], we will follow these steps:

### Step 1: Rewrite the square root in exponential form
We know that \(\sqrt{6} = 6^{1/2}\). Therefore, we can rewrite the expression as:
\[ \left[ \frac{(6^{1/2})^5 \times (6^{1/2})^{-3}}{(6^{1/2})^{-2}} \right]^{\frac{3}{2}} \]

### Step 2: Apply the power of a power property
Using the property \((a^m)^n = a^{m \cdot n}\), we can simplify the powers:
\[ = \left[ \frac{6^{5/2} \times 6^{-3/2}}{6^{-1}} \right]^{\frac{3}{2}} \]

### Step 3: Combine the powers in the numerator
When multiplying powers with the same base, we add the exponents:
\[ = \left[ \frac{6^{5/2 - 3/2}}{6^{-1}} \right]^{\frac{3}{2}} = \left[ \frac{6^{2/2}}{6^{-1}} \right]^{\frac{3}{2}} = \left[ \frac{6^1}{6^{-1}} \right]^{\frac{3}{2}} \]

### Step 4: Simplify the fraction
When dividing powers with the same base, we subtract the exponents:
\[ = \left[ 6^{1 - (-1)} \right]^{\frac{3}{2}} = \left[ 6^{1 + 1} \right]^{\frac{3}{2}} = \left[ 6^2 \right]^{\frac{3}{2}} \]

### Step 5: Apply the power of a power property again
Now we simplify further:
\[ = 6^{2 \cdot \frac{3}{2}} = 6^3 \]

### Step 6: Calculate the final value
Now we compute \(6^3\):
\[ 6^3 = 216 \]

Thus, the value of the expression is \(\boxed{216}\).

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