Find the value of `(8^4+8^2)^(1/2)`

Options

A. 84
B. `8sqrt(77)`
C. 72
D. `8sqrt(65)`

Explanation

To solve the expression \((8^4 + 8^2)^{1/2}\), we can follow these steps:

### Step 1: Simplify the expression inside the parentheses
We start with the expression:
\[ 8^4 + 8^2 \]
We can factor out \(8^2\) from both terms:
\[ 8^4 + 8^2 = 8^2(8^2 + 1) \]

### Step 2: Calculate \(8^2\)
Now, we calculate \(8^2\):
\[ 8^2 = 64 \]
So, we can substitute this back into the expression:
\[ 8^4 + 8^2 = 64(8^2 + 1) = 64(64 + 1) = 64 \times 65 \]

### Step 3: Substitute back into the original expression
Now we substitute this back into the original expression:
\[ (8^4 + 8^2)^{1/2} = (64 \times 65)^{1/2} \]

### Step 4: Use the property of exponents
Using the property of exponents, we can separate the square root:
\[ (64 \times 65)^{1/2} = (64^{1/2}) \times (65^{1/2}) \]

### Step 5: Calculate \(64^{1/2}\)
Now we calculate \(64^{1/2}\):
\[ 64^{1/2} = 8 \]
So, we have:
\[ (8 \times 65^{1/2}) \]

### Step 6: Final expression
Thus, the final expression becomes:
\[ 8 \sqrt{65} \]

### Conclusion
The value of \((8^4 + 8^2)^{1/2}\) is:
\[ 8 \sqrt{65} \]