Simplified value of: `{(4^(m+1/4) xx sqrt(2.2^(m)))/(2sqrt(2^(-m)))}^(1//m)` is:

Options

A. 8
B. 4
C. 16
D. 2

Answer

To simplify the expression \(\left(\frac{4^{m + \frac{1}{4}} \times \sqrt{2 \cdot 2^{m}}}{2 \sqrt{2^{-m}}}\right)^{\frac{1}{m}}\), we can follow these steps:

### Step 1: Rewrite the base in terms of powers of 2
We know that \(4 = 2^2\). Therefore, we can rewrite \(4^{m + \frac{1}{4}}\) as:
\[ 4^{m + \frac{1}{4}} = (2^2)^{m + \frac{1}{4}} = 2^{2(m + \frac{1}{4})} = 2^{2m + \frac{1}{2}} \]

### Step 2: Simplify the square root term
Next, we simplify \(\sqrt{2 \cdot 2^{m}}\):
\[ \sqrt{2 \cdot 2^{m}} = \sqrt{2^{1 + m}} = 2^{\frac{1 + m}{2}} \]

### Step 3: Simplify the denominator
Now, we simplify the denominator \(2 \sqrt{2^{-m}}\):
\[ 2 \sqrt{2^{-m}} = 2 \cdot 2^{-\frac{m}{2}} = 2^{1 - \frac{m}{2}} \]

### Step 4: Combine the terms in the numerator
Now we can combine the terms in the numerator:
\[ \text{Numerator} = 2^{2m + \frac{1}{2}} \times 2^{\frac{1 + m}{2}} = 2^{(2m + \frac{1}{2}) + (\frac{1 + m}{2})} \]
Calculating the exponent:
\[ 2m + \frac{1}{2} + \frac{1 + m}{2} = 2m + \frac{1}{2} + \frac{1}{2} + \frac{m}{2} = 2m + \frac{1 + 1}{2} + \frac{m}{2} = 2m + 1 + \frac{m}{2} = \frac{4m + 2 + m}{2} = \frac{5m + 2}{2} \]
Thus, the numerator simplifies to:
\[ \text{Numerator} = 2^{\frac{5m + 2}{2}} \]

### Step 5: Combine the entire fraction
Now we can write the entire expression:
\[ \frac{2^{\frac{5m + 2}{2}}}{2^{1 - \frac{m}{2}}} = 2^{\frac{5m + 2}{2} - (1 - \frac{m}{2})} \]
Calculating the exponent:
\[ \frac{5m + 2}{2} - 1 + \frac{m}{2} = \frac{5m + 2 - 2 + m}{2} = \frac{6m}{2} = 3m \]
So, the entire fraction simplifies to:
\[ 2^{3m} \]

### Step 6: Apply the exponent \(\frac{1}{m}\)
Now we apply the exponent \(\frac{1}{m}\):
\[ (2^{3m})^{\frac{1}{m}} = 2^{3} \]

### Step 7: Final calculation
Finally, we calculate \(2^3\):
\[ 2^3 = 8 \]

### Conclusion
Thus, the simplified value of the expression is:
\[ \boxed{8} \]