Sure, let's proceed step by step to simplify the expression [tex]\(\sqrt[5]{4} \cdot \sqrt{2}\)[/tex].
1. First, we'll rewrite each radical expression using rational exponents.
[tex]\[
\sqrt[5]{4} \text{ can be written as } 4^{\frac{1}{5}}
\][/tex]
Since [tex]\(4 = 2^2\)[/tex], this becomes:
[tex]\[
4^{\frac{1}{5}} = (2^2)^{\frac{1}{5}} = 2^{\frac{2}{5}}
\][/tex]
Similarly,
[tex]\[
\sqrt{2} \text{ can be written as } 2^{\frac{1}{2}}
\][/tex]
2. Now, we rewrite the original expression [tex]\(\sqrt[5]{4} \cdot \sqrt{2}\)[/tex] using these rational exponents:
[tex]\[
2^{\frac{2}{5}} \cdot 2^{\frac{1}{2}}
\][/tex]
3. When multiplying like bases (in this case, base 2), we add the exponents:
[tex]\[
2^{\frac{2}{5}} \cdot 2^{\frac{1}{2}} = 2^{\frac{2}{5} + \frac{1}{2}}
\][/tex]
4. To add the exponents [tex]\(\frac{2}{5}\)[/tex] and [tex]\(\frac{1}{2}\)[/tex], we need a common denominator. The least common denominator (LCD) of 5 and 2 is:
[tex]\[
10
\][/tex]
So, the least common denominator of the exponents [tex]\(\frac{2}{5}\)[/tex] and [tex]\(\frac{1}{2}\)[/tex] is [tex]\(\boxed{10}\)[/tex].
