To solve the given expression [tex]\(13^{\frac{1}{5}} \cdot 17^{\frac{1}{5}}\)[/tex], we will follow a step-by-step approach to simplify and evaluate it.
1. Identify the base and exponent of each term:
- The first term is [tex]\(13\)[/tex] raised to the power of [tex]\(\frac{1}{5}\)[/tex].
- The second term is [tex]\(17\)[/tex] raised to the power of [tex]\(\frac{1}{5}\)[/tex].
2. Evaluate [tex]\(13^{\frac{1}{5}}\)[/tex]:
By finding the fifth root of [tex]\(13\)[/tex], we get:
[tex]\[
13^{\frac{1}{5}} \approx 1.6702776523348104
\][/tex]
3. Evaluate [tex]\(17^{\frac{1}{5}}\)[/tex]:
By finding the fifth root of [tex]\(17\)[/tex], we get:
[tex]\[
17^{\frac{1}{5}} \approx 1.762340347832317
\][/tex]
4. Multiply the results of the two evaluations:
[tex]\[
1.6702776523348104 \cdot 1.762340347832317 \approx 2.9435976987922756
\][/tex]
Therefore, the expression [tex]\(13^{\frac{1}{5}} \cdot 17^{\frac{1}{5}}\)[/tex] evaluates to approximately
[tex]\[
\boxed{2.9435976987922756}
\][/tex]
