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

Options

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

Answer

To solve the expression \( 8 - \left( \frac{4^{(9/4)} \sqrt{(2.2)^2}}{2 \sqrt{2^{-2}}} \right)^{1/2} \), we will break it down step by step.

### Step 1: Simplify the expression inside the parentheses
We start with the expression:
\[ \frac{4^{(9/4)} \sqrt{(2.2)^2}}{2 \sqrt{2^{-2}}} \]

First, simplify \( \sqrt{(2.2)^2} \):
\[ \sqrt{(2.2)^2} = 2.2 \]

Next, simplify \( \sqrt{2^{-2}} \):
\[ \sqrt{2^{-2}} = \frac{1}{\sqrt{2^2}} = \frac{1}{2} \]

Now substitute these simplifications back into the expression:
\[ \frac{4^{(9/4)} \cdot 2.2}{2 \cdot \frac{1}{2}} = \frac{4^{(9/4)} \cdot 2.2}{1} = 4^{(9/4)} \cdot 2.2 \]

### Step 2: Simplify \( 4^{(9/4)} \)
Recall that \( 4 = 2^2 \), so:
\[ 4^{(9/4)} = (2^2)^{(9/4)} = 2^{(2 \cdot \frac{9}{4})} = 2^{(18/4)} = 2^{(9/2)} \]

### Step 3: Substitute back into the expression
Now we have:
\[ 4^{(9/4)} \cdot 2.2 = 2^{(9/2)} \cdot 2.2 \]

### Step 4: Rewrite \( 2.2 \) in terms of powers of 2
We can express \( 2.2 \) as:
\[ 2.2 = 2^{1} \cdot 2^{0.3219} \text{ (approximately)} \]
However, for simplicity, we can just keep it as \( 2.2 \) for now.

### Step 5: Combine the terms
Now we combine:
\[ 2^{(9/2)} \cdot 2.2 = 2^{(9/2)} \cdot 2^{1} = 2^{(9/2 + 1)} = 2^{(9/2 + 2/2)} = 2^{(11/2)} \]

### Step 6: Substitute back into the square root
Now we substitute this back into the square root:
\[ \left( 2^{(11/2)} \right)^{1/2} = 2^{(11/4)} \]

### Step 7: Substitute into the original expression
Now we substitute back into the original expression:
\[ 8 - 2^{(11/4)} \]

### Step 8: Simplify \( 8 \)
Recall that \( 8 = 2^3 \), so:
\[ 8 - 2^{(11/4)} = 2^{3} - 2^{(11/4)} \]

### Step 9: Find a common base
To combine these, we can express \( 2^3 \) as \( 2^{(12/4)} \):
\[ 2^{(12/4)} - 2^{(11/4)} = 2^{(12/4)} - 2^{(11/4)} = 2^{(11/4)}(2^{(1/4)} - 1) \]

### Step 10: Evaluate the final expression
Since \( 2^{(1/4)} - 1 \) is a positive number, we can conclude that:
\[ 2^{(11/4)}(2^{(1/4)} - 1) \neq 0 \]
Thus, \( 8 - 2^{(11/4)} \) does not equal zero.

### Final Result
The final result is:
\[ 8 - 2^{(11/4)} \text{ (exact value)} \]