To determine which of the provided options is equivalent to the expression [tex]\(\sqrt[7]{100^6}\)[/tex], follow these steps:
1. Rewrite the Radical Expression as a Fractional Exponent:
[tex]\[
\sqrt[7]{100^6} = 100^{\frac{6}{7}}
\][/tex]
2. Simplify [tex]\(\mathbf{100^{\frac{6}{7}}}\)[/tex] by Expressing [tex]\(\mathbf{100}\)[/tex] as [tex]\(\mathbf{10^2}\)[/tex]:
[tex]\[
100 = 10^2
\][/tex]
Therefore,
[tex]\[
100^{\frac{6}{7}} = \left(10^2\right)^{\frac{6}{7}}
\][/tex]
3. Use the Power of a Power Property to Combine the Exponents:
[tex]\[
\left(10^2\right)^{\frac{6}{7}} = 10^{2 \cdot \frac{6}{7}} = 10^{\frac{12}{7}}
\][/tex]
4. Recognize That [tex]\(\mathbf{10^{\frac{12}{7}}}\)[/tex] Equals One of the Given Options:
[tex]\[
\mathbf{D. \ 10^{\frac{6}{7}}}
\][/tex]
Therefore, the equivalent expression is:
[tex]\[
\boxed{10^{\frac{6}{7}}}
\][/tex]
So, the correct answer is [tex]\(\mathbf{D: \ 10^{\frac{6}{7}}}\)[/tex].
