Complete the steps to simplify [tex]$\sqrt[5]{4} \cdot \sqrt{2}$[/tex].

Rewrite using rational exponents.

[tex]
\begin{aligned}
&2^{\frac{2}{5}} \cdot 2^{\frac{1}{2}}
\end{aligned}
[/tex]

What is the least common denominator of the exponents?

[tex]\square[/tex]



Answer :

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].