`2^(0.2)xx64xx8^(1.3)xx4^(0.2)=8^(?)`

Options

A. 2.7
B. 2.5
C. 3.7
D. None of these

Explanation

To solve the equation \( 2^{0.2} \times 64 \times 8^{1.3} \times 4^{0.2} = 8^{?} \), we will simplify the left-hand side step by step and find the value of the question mark.

### Step 1: Rewrite all terms as powers of 2
- We know that \( 64 = 2^6 \) and \( 8 = 2^3 \).
- Also, \( 4 = 2^2 \).

Thus, we can rewrite the equation as:
\[ 2^{0.2} \times 2^6 \times (2^3)^{1.3} \times (2^2)^{0.2} \]

### Step 2: Simplify the powers
- Now we simplify \( (2^3)^{1.3} = 2^{3 \times 1.3} = 2^{3.9} \).
- And \( (2^2)^{0.2} = 2^{2 \times 0.2} = 2^{0.4} \).

Now, substituting these back, we have:
\[ 2^{0.2} \times 2^6 \times 2^{3.9} \times 2^{0.4} \]

### Step 3: Combine the exponents
- We can combine the exponents since they have the same base:
\[ 2^{0.2 + 6 + 3.9 + 0.4} \]

### Step 4: Calculate the sum of the exponents
- Now, we calculate the sum:
\[ 0.2 + 6 = 6.2 \]
\[ 6.2 + 3.9 = 10.1 \]
\[ 10.1 + 0.4 = 10.5 \]

So, we have:
\[ 2^{10.5} \]

### Step 5: Rewrite the right-hand side
- The right-hand side is \( 8^{?} \), which we can also express as:
\[ (2^3)^{?} = 2^{3?} \]

### Step 6: Set the exponents equal
- Now, we can set the exponents equal to each other:
\[ 10.5 = 3? \]

### Step 7: Solve for ?
- To find \( ? \), we divide both sides by 3:
\[ ? = \frac{10.5}{3} = 3.5 \]

### Final Answer
Thus, the value of the question mark is:
\[ \boxed{3.5} \]