Value of `(4096) ^(((1)/(sqrt3))^((sqrt2) ^(2)))` is

Options

A. 8
B. 6
C. 24
D. 32

Answer

To solve the expression \( (4096)^{\left(\frac{1}{\sqrt{3}}\right)^{\left(\sqrt{2}\right)^2}} \), we can follow these steps:

### Step 1: Simplify the exponent
First, we simplify the exponent \( \left(\frac{1}{\sqrt{3}}\right)^{\left(\sqrt{2}\right)^2} \).

Since \( \left(\sqrt{2}\right)^2 = 2 \), we can rewrite the exponent as:
\[ \left(\frac{1}{\sqrt{3}}\right)^{2} = \frac{1}{3} \]

### Step 2: Rewrite the original expression
Now we can rewrite the original expression using the simplified exponent:
\[ (4096)^{\frac{1}{3}} \]

### Step 3: Express 4096 as a power of 2
Next, we need to express 4096 as a power of 2. We know that:
\[ 4096 = 2^{12} \]

### Step 4: Substitute back into the expression
Now we substitute \( 4096 \) in the expression:
\[ (2^{12})^{\frac{1}{3}} \]

### Step 5: Apply the power of a power property
Using the property of exponents \( (a^m)^n = a^{m \cdot n} \), we can simplify further:
\[ 2^{12 \cdot \frac{1}{3}} = 2^{4} \]

### Step 6: Calculate the final value
Now we calculate \( 2^4 \):
\[ 2^4 = 16 \]

Thus, the value of \( (4096)^{\left(\frac{1}{\sqrt{3}}\right)^{\left(\sqrt{2}\right)^2}} \) is \( \boxed{16} \).

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