To determine which expression is equivalent to [tex]\(\frac{\sqrt{10}}{\sqrt[4]{8}}\)[/tex], we will follow a series of steps to simplify and compare the expressions.
1. Understand the Given Problem:
We have the expression [tex]\(\frac{\sqrt{10}}{\sqrt[4]{8}}\)[/tex].
2. Rewrite the Denominator with Exponents:
[tex]\(\sqrt[4]{8}\)[/tex] can be rewritten using fractional exponents. We know [tex]\(8 = 2^3\)[/tex], so:
[tex]\[
\sqrt[4]{8} = (8)^{\frac{1}{4}} = (2^3)^{\frac{1}{4}} = 2^{3/4}
\][/tex]
3. Rewrite the Entire Expression:
Using this result, we can rewrite the original expression:
[tex]\[
\frac{\sqrt{10}}{\sqrt[4]{8}} = \frac{\sqrt{10}}{2^{3/4}}
\][/tex]
4. Simplify Further Using Algebraic Rules:
Using properties of exponents and radicals, we know:
[tex]\[
\frac{\sqrt{a}}{b^c} = \frac{\sqrt{a}}{b^{1/2}} \cdot \frac{1}{b^{c-1/2}}
\][/tex]
However, in this case, we notice the numerical simplification isn’t straightforward.
5. Compare With Given Options:
We can examine the provided choices to find a match:
a. [tex]\(\frac{\sqrt[4]{200}}{2}\)[/tex]
b. [tex]\(\frac{\sqrt[4]{20}}{2}\)[/tex]
c. [tex]\(\frac{2 \sqrt{5}}{5}\)[/tex]
d. [tex]\(\frac{100}{8}\)[/tex]
6. Substitute and Simplify Each Option:
- For [tex]\(\frac{\sqrt[4]{200}}{2}\)[/tex]: This simplifies to [tex]\(\frac{(200)^{1/4}}{2}\)[/tex], but does not match our target expression.
- For [tex]\(\frac{\sqrt[4]{20}}{2}\)[/tex]: This simplifies to [tex]\(\frac{(20)^{1/4}}{2}\)[/tex], but again does not match our target expression.
- For [tex]\(\frac{2 \sqrt{5}}{5}\)[/tex]:
[tex]\[
\frac{2 \sqrt{5}}{5} = 2 \cdot \frac{\sqrt{5}}{5}
\][/tex]
This does not look like our simplified form [tex]\(\frac{\sqrt{10}}{2^{3/4}}\)[/tex].
- For [tex]\(\frac{100}{8}\)[/tex]:
[tex]\[
\frac{100}{8} = 12.5
\][/tex]
This is a simple division which clearly does not match our radical form.
Referring back to the result, [tex]\(\frac{\sqrt{10}}{2^{3/4}} = \frac{2^{3/4} \sqrt{5}}{2}\)[/tex] is the closest to the simplified form, validating:
[tex]\[
\boxed{\text{None of the provided choices match the simplified form}}
\][/tex]
