`[8-((4^((9)/(4))sqrt(2.2^(2)))/(2sqrt(2^(-2))))^((1)/(2))]` is equal to

Options

A. `32`
B. `8`
C. `1`
D. `0`

Answer

To solve the expression \( 8 - \left( \frac{4^{\frac{9}{4}} \sqrt{2} \cdot 2^{2}}{2 \sqrt{2^{-2}}} \right)^{\frac{1}{2}} \), we will simplify it step by step.

### Step 1: Simplify the components inside the parentheses

First, we rewrite the expression inside the parentheses:

\[ \frac{4^{\frac{9}{4}} \sqrt{2} \cdot 2^{2}}{2 \sqrt{2^{-2}}} \]

We know that \( 4 = 2^2 \), so we can rewrite \( 4^{\frac{9}{4}} \) as:

\[ (2^2)^{\frac{9}{4}} = 2^{2 \cdot \frac{9}{4}} = 2^{\frac{18}{4}} = 2^{\frac{9}{2}} \]

Now, we can rewrite \( \sqrt{2} \) as \( 2^{\frac{1}{2}} \) and \( 2^{2} \) as \( 2^{2} \).

So, the numerator becomes:

\[ 2^{\frac{9}{2}} \cdot 2^{\frac{1}{2}} \cdot 2^{2} = 2^{\frac{9}{2} + \frac{1}{2} + 2} = 2^{\frac{9}{2} + \frac{1}{2} + \frac{4}{2}} = 2^{\frac{14}{2}} = 2^{7} \]

### Step 2: Simplify the denominator

Now, we simplify the denominator:

\[ 2 \sqrt{2^{-2}} = 2 \cdot 2^{-1} = 2^{1} \cdot 2^{-1} = 2^{0} = 1 \]

### Step 3: Combine the results

Now we can combine the results from the numerator and denominator:

\[ \frac{2^{7}}{1} = 2^{7} \]

### Step 4: Apply the square root

Now we apply the square root to the entire expression:

\[ \left( 2^{7} \right)^{\frac{1}{2}} = 2^{\frac{7}{2}} \]

### Step 5: Substitute back into the original expression

Now we substitute this back into the original expression:

\[ 8 - 2^{\frac{7}{2}} \]

### Step 6: Rewrite 8 in terms of powers of 2

We know that \( 8 = 2^{3} \), so we rewrite the expression:

\[ 2^{3} - 2^{\frac{7}{2}} \]

### Step 7: Find a common base

To combine these, we can express \( 2^{3} \) as \( 2^{\frac{6}{2}} \):

\[ 2^{\frac{6}{2}} - 2^{\frac{7}{2}} = 2^{\frac{6}{2}} - 2^{\frac{7}{2}} = 2^{\frac{6}{2}}(1 - 2^{\frac{1}{2}}) \]

### Step 8: Simplify the expression

This simplifies to:

\[ 2^{\frac{6}{2}}(1 - \sqrt{2}) = 2^{3}(1 - \sqrt{2}) \]

### Final Result

Thus, the final answer is:

\[ 8(1 - \sqrt{2}) \]