To simplify the expression [tex]\(\sqrt{64 x^{16}}\)[/tex], let's break it down into smaller steps.
### Step 1: Understand the expression inside the square root
The given expression is [tex]\(\sqrt{64 x^{16}}\)[/tex]. This can be separated into two parts under the square root:
[tex]\[ \sqrt{64 \cdot x^{16}} \][/tex]
### Step 2: Apply the square root to each part separately
The property of square roots allows us to split the square root into the product of the square roots of each factor:
[tex]\[ \sqrt{64 \cdot x^{16}} = \sqrt{64} \cdot \sqrt{x^{16}} \][/tex]
### Step 3: Compute the square root of 64
The square root of 64 is:
[tex]\[ \sqrt{64} = 8 \][/tex]
### Step 4: Simplify the square root of [tex]\(x^{16}\)[/tex]
For the term [tex]\( \sqrt{x^{16}} \)[/tex], we use the property of exponents:
[tex]\[ \sqrt{x^{16}} = (x^{16})^{1/2} \][/tex]
By multiplying the powers:
[tex]\[ (x^{16})^{1/2} = x^{16 \cdot \frac{1}{2}} = x^{8} \][/tex]
### Step 5: Combine the results
Now, we multiply the results of the square roots of each part:
[tex]\[ 8 \cdot x^{8} \][/tex]
Therefore, the simplified form of [tex]\(\sqrt{64 x^{16}}\)[/tex] is:
[tex]\[ 8 x^{8} \][/tex]
So, the correct answer is:
[tex]\[ \boxed{8 x^8} \][/tex]
