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Sagot :
Sure, let's solve the given mathematical expression step by step. The expression we're working with is:
[tex]\[ \left(\frac{4^{m+\frac{1}{4}} \times \sqrt{2.2^m}}{2 \cdot \sqrt{2^{-m}}}\right)^{\frac{1}{m}} \][/tex]
### Step 1: Simplify the Expression Inside the Parentheses
First, let's focus on simplifying the fraction inside the parentheses.
#### Numerator:
[tex]\[ 4^{m+\frac{1}{4}} \times \sqrt{2.2^m} \][/tex]
We know that:
[tex]\[ 4 = 2^2 \][/tex]
So,
[tex]\[ 4^{m+\frac{1}{4}} = (2^2)^{m+\frac{1}{4}} = 2^{2(m+\frac{1}{4})} = 2^{2m + \frac{1}{2}} \][/tex]
Also,
[tex]\[ \sqrt{2.2^m} = (2.2^m)^{\frac{1}{2}} = 2.2^{\frac{m}{2}} \][/tex]
So the numerator becomes:
[tex]\[ 2^{2m + \frac{1}{2}} \times 2.2^{\frac{m}{2}} \][/tex]
#### Denominator:
[tex]\[ 2 \cdot \sqrt{2^{-m}} \][/tex]
We know that:
[tex]\[ \sqrt{2^{-m}} = (2^{-m})^{\frac{1}{2}} = 2^{-\frac{m}{2}} \][/tex]
So the denominator becomes:
[tex]\[ 2 \cdot 2^{-\frac{m}{2}} \][/tex]
The entire fraction now is:
[tex]\[ \frac{2^{2m + \frac{1}{2}} \times 2.2^{\frac{m}{2}}}{2 \cdot 2^{-\frac{m}{2}}} \][/tex]
### Step 2: Simplify the Expression Further
Combine the terms in the denominator:
[tex]\[ 2 \cdot 2^{-\frac{m}{2}} = 2^{1 - \frac{m}{2}} \][/tex]
So, the entire expression becomes:
[tex]\[ \frac{2^{2m + \frac{1}{2}} \times 2.2^{\frac{m}{2}}}{2^{1 - \frac{m}{2}}} \][/tex]
Subtracting the exponents in the denominator from the exponents in the numerator gives:
[tex]\[ 2^{2m + \frac{1}{2} - (1 - \frac{m}{2})} \times 2.2^{\frac{m}{2}} \][/tex]
[tex]\[ = 2^{2m + \frac{1}{2} - 1 + \frac{m}{2}} \times 2.2^{\frac{m}{2}} \][/tex]
[tex]\[ = 2^{2m - \frac{1}{2} + \frac{m}{2}} \times 2.2^{\frac{m}{2}} \][/tex]
[tex]\[ = 2^{2m - \frac{1}{2} + \frac{m}{2}} \times 2.2^{\frac{m}{2}} \][/tex]
[tex]\[ = 2^{2m - \frac{1}{2} + \frac{m}{2}} \][/tex]
### Step 3: Raise the Expression to the Power of [tex]\(\frac{1}{m}\)[/tex]
Now, we want to raise the simplified expression to the power of [tex]\(\frac{1}{m}\)[/tex]:
[tex]\[ \left(2^{2m - \frac{1}{2} + \frac{m}{2}} \times 2.2^{\frac{m}{2}}\right)^{\frac{1}{m}} \][/tex]
This distributes as:
[tex]\[ \left(2^{2m - \frac{1}{2} + \frac{m}{2}}\right)^{\frac{1}{m}} \times \left(2.2^{\frac{m}{2}}\right)^{\frac{1}{m}} \][/tex]
Breaking it down,
[tex]\[ \left(2^{\frac{5m}{2} - \frac{1}{2}}\right)^{\frac{1}{m}} = 2^{\left(\frac{5m}{2} - \frac{1}{2}\right) \cdot \frac{1}{m}} = 2^{\frac{5}{2} - \frac{1}{2m}} \][/tex]
For the second term:
[tex]\[ \left(2.2^{\frac{m}{2}}\right)^{\frac{1}{m}} = 2.2^{\frac{1}{2}} \][/tex]
Therefore, putting them together:
[tex]\[ 2^{2 - \frac{1}{2m}} \times 2.2^{\frac{1}{2}} \][/tex]
### Final Answer
The fully simplified expression is:
[tex]\[ \boxed{2^{2 - \frac{1}{2m}} \times 2.2^{\frac{1}{2}}} \][/tex]
[tex]\[ \left(\frac{4^{m+\frac{1}{4}} \times \sqrt{2.2^m}}{2 \cdot \sqrt{2^{-m}}}\right)^{\frac{1}{m}} \][/tex]
### Step 1: Simplify the Expression Inside the Parentheses
First, let's focus on simplifying the fraction inside the parentheses.
#### Numerator:
[tex]\[ 4^{m+\frac{1}{4}} \times \sqrt{2.2^m} \][/tex]
We know that:
[tex]\[ 4 = 2^2 \][/tex]
So,
[tex]\[ 4^{m+\frac{1}{4}} = (2^2)^{m+\frac{1}{4}} = 2^{2(m+\frac{1}{4})} = 2^{2m + \frac{1}{2}} \][/tex]
Also,
[tex]\[ \sqrt{2.2^m} = (2.2^m)^{\frac{1}{2}} = 2.2^{\frac{m}{2}} \][/tex]
So the numerator becomes:
[tex]\[ 2^{2m + \frac{1}{2}} \times 2.2^{\frac{m}{2}} \][/tex]
#### Denominator:
[tex]\[ 2 \cdot \sqrt{2^{-m}} \][/tex]
We know that:
[tex]\[ \sqrt{2^{-m}} = (2^{-m})^{\frac{1}{2}} = 2^{-\frac{m}{2}} \][/tex]
So the denominator becomes:
[tex]\[ 2 \cdot 2^{-\frac{m}{2}} \][/tex]
The entire fraction now is:
[tex]\[ \frac{2^{2m + \frac{1}{2}} \times 2.2^{\frac{m}{2}}}{2 \cdot 2^{-\frac{m}{2}}} \][/tex]
### Step 2: Simplify the Expression Further
Combine the terms in the denominator:
[tex]\[ 2 \cdot 2^{-\frac{m}{2}} = 2^{1 - \frac{m}{2}} \][/tex]
So, the entire expression becomes:
[tex]\[ \frac{2^{2m + \frac{1}{2}} \times 2.2^{\frac{m}{2}}}{2^{1 - \frac{m}{2}}} \][/tex]
Subtracting the exponents in the denominator from the exponents in the numerator gives:
[tex]\[ 2^{2m + \frac{1}{2} - (1 - \frac{m}{2})} \times 2.2^{\frac{m}{2}} \][/tex]
[tex]\[ = 2^{2m + \frac{1}{2} - 1 + \frac{m}{2}} \times 2.2^{\frac{m}{2}} \][/tex]
[tex]\[ = 2^{2m - \frac{1}{2} + \frac{m}{2}} \times 2.2^{\frac{m}{2}} \][/tex]
[tex]\[ = 2^{2m - \frac{1}{2} + \frac{m}{2}} \times 2.2^{\frac{m}{2}} \][/tex]
[tex]\[ = 2^{2m - \frac{1}{2} + \frac{m}{2}} \][/tex]
### Step 3: Raise the Expression to the Power of [tex]\(\frac{1}{m}\)[/tex]
Now, we want to raise the simplified expression to the power of [tex]\(\frac{1}{m}\)[/tex]:
[tex]\[ \left(2^{2m - \frac{1}{2} + \frac{m}{2}} \times 2.2^{\frac{m}{2}}\right)^{\frac{1}{m}} \][/tex]
This distributes as:
[tex]\[ \left(2^{2m - \frac{1}{2} + \frac{m}{2}}\right)^{\frac{1}{m}} \times \left(2.2^{\frac{m}{2}}\right)^{\frac{1}{m}} \][/tex]
Breaking it down,
[tex]\[ \left(2^{\frac{5m}{2} - \frac{1}{2}}\right)^{\frac{1}{m}} = 2^{\left(\frac{5m}{2} - \frac{1}{2}\right) \cdot \frac{1}{m}} = 2^{\frac{5}{2} - \frac{1}{2m}} \][/tex]
For the second term:
[tex]\[ \left(2.2^{\frac{m}{2}}\right)^{\frac{1}{m}} = 2.2^{\frac{1}{2}} \][/tex]
Therefore, putting them together:
[tex]\[ 2^{2 - \frac{1}{2m}} \times 2.2^{\frac{1}{2}} \][/tex]
### Final Answer
The fully simplified expression is:
[tex]\[ \boxed{2^{2 - \frac{1}{2m}} \times 2.2^{\frac{1}{2}}} \][/tex]
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