To simplify the expression [tex]\(\sqrt{10,000 x^{64}}\)[/tex], let's proceed step by step:
1. Recognize the Expression: The given expression is [tex]\(\sqrt{10,000 x^{64}}\)[/tex].
2. Factor the Expression:
The square root function is distributive over multiplication for non-negative values, meaning:
[tex]\[
\sqrt{A \cdot B} = \sqrt{A} \cdot \sqrt{B}
\][/tex]
Applying this property to [tex]\(\sqrt{10,000 x^{64}}\)[/tex]:
[tex]\[
\sqrt{10,000 x^{64}} = \sqrt{10,000} \cdot \sqrt{x^{64}}
\][/tex]
3. Simplify Each Component:
- First, simplify [tex]\(\sqrt{10,000}\)[/tex]:
[tex]\[
\sqrt{10,000} = \sqrt{100^2} = 100
\][/tex]
- Next, simplify [tex]\(\sqrt{x^{64}}\)[/tex]:
Note that the square root of [tex]\(x^{64}\)[/tex] is:
[tex]\[
\sqrt{x^{64}} = (x^{64})^{\frac{1}{2}} = x^{64 \cdot \frac{1}{2}} = x^{32}
\][/tex]
4. Combine the Results:
Putting these simplified parts together, we get:
[tex]\[
\sqrt{10,000 x^{64}} = 100 \cdot x^{32}
\][/tex]
Hence, the simplified form of [tex]\(\sqrt{10,000 x^{64}}\)[/tex] is:
[tex]\[
100 x^{32}
\][/tex]
Thus, the correct answer is:
[tex]\[
\boxed{100 x^{32}}
\][/tex]
