To simplify the expression [tex]\(\frac{\sqrt[3]{7}}{\sqrt[5]{7}}\)[/tex], we need to handle the exponents involved and simplify accordingly.
First, let's express the radicals in exponent form:
[tex]\[
\sqrt[3]{7} = 7^{\frac{1}{3}}
\][/tex]
[tex]\[
\sqrt[5]{7} = 7^{\frac{1}{5}}
\][/tex]
Now, let's rewrite the given expression [tex]\(\frac{\sqrt[3]{7}}{\sqrt[5]{7}}\)[/tex] using these exponents:
[tex]\[
\frac{7^{\frac{1}{3}}}{7^{\frac{1}{5}}}
\][/tex]
When we divide powers with the same base, we subtract the exponents:
[tex]\[
7^{\frac{1}{3} - \frac{1}{5}}
\][/tex]
To subtract these fractions, we need a common denominator. The least common multiple of 3 and 5 is 15. So we convert both fractions to have a denominator of 15:
[tex]\[
\frac{1}{3} = \frac{5}{15}
\][/tex]
[tex]\[
\frac{1}{5} = \frac{3}{15}
\][/tex]
Now, subtract the exponents:
[tex]\[
\frac{5}{15} - \frac{3}{15} = \frac{2}{15}
\][/tex]
This gives us the simplified exponent:
[tex]\[
7^{\frac{2}{15}}
\][/tex]
Thus, the simplified form of the expression [tex]\(\frac{\sqrt[3]{7}}{\sqrt[5]{7}}\)[/tex] is:
[tex]\[
7^{\frac{2}{15}}
\][/tex]
Among the given choices:
1. [tex]\(7^{\frac{1}{5}}\)[/tex]
2. [tex]\(7^{\frac{8}{15}}\)[/tex]
3. [tex]\(7^{\frac{5}{3}}\)[/tex]
4. [tex]\(7^{\frac{2}{15}}\)[/tex]
The correct answer is:
[tex]\[
7^{\frac{2}{15}}
\][/tex]
