To simplify the expression [tex]\(\sqrt{10,000 x^{64}}\)[/tex], we need to handle the square root of each part of the expression separately.
1. Simplify the numerical part: [tex]\(\sqrt{10,000}\)[/tex]
We know that [tex]\(10,000\)[/tex] can be written as [tex]\(10,000 = 100^2\)[/tex], which implies that:
[tex]\[
\sqrt{10,000} = \sqrt{100^2} = 100
\][/tex]
2. Simplify the variable part: [tex]\(\sqrt{x^{64}}\)[/tex]
Using the property of exponents, [tex]\(\sqrt{x^{64}}\)[/tex] can be rewritten as [tex]\(x^{64/2}\)[/tex], which simplifies to:
[tex]\[
\sqrt{x^{64}} = x^{64/2} = x^{32}
\][/tex]
Now, combining the simplified numerical and variable parts, we get:
[tex]\[
\sqrt{10,000 x^{64}} = 100 \cdot x^{32}
\][/tex]
Thus, the simplified form of [tex]\(\sqrt{10,000 x^{64}}\)[/tex] is:
[tex]\[
\boxed{100 x^{32}}
\][/tex]
