To simplify the given expression [tex]\(\frac{\sqrt[5]{a^4}}{\sqrt[3]{a^2}}\)[/tex], follow these steps:
1. Convert the Radicals to Fractional Exponents:
- [tex]\(\sqrt[5]{a^4}\)[/tex] can be written as [tex]\(a^{\frac{4}{5}}\)[/tex].
- [tex]\(\sqrt[3]{a^2}\)[/tex] can be written as [tex]\(a^{\frac{2}{3}}\)[/tex].
So the given expression becomes:
[tex]\[
\frac{a^{\frac{4}{5}}}{a^{\frac{2}{3}}}
\][/tex]
2. Apply the Division Property of Exponents:
When dividing expressions with the same base, subtract the exponents:
[tex]\[
a^{\frac{4}{5}} \div a^{\frac{2}{3}} = a^{\left(\frac{4}{5} - \frac{2}{3}\right)}
\][/tex]
3. Find a Common Denominator to Subtract the Exponents:
The denominators are 5 and 3. The least common multiple of 5 and 3 is 15.
Rewrite the exponents with a common denominator:
[tex]\[
\frac{4}{5} = \frac{4 \times 3}{5 \times 3} = \frac{12}{15}
\][/tex]
[tex]\[
\frac{2}{3} = \frac{2 \times 5}{3 \times 5} = \frac{10}{15}
\][/tex]
Now, subtract the exponents:
[tex]\[
\frac{12}{15} - \frac{10}{15} = \frac{12 - 10}{15} = \frac{2}{15}
\][/tex]
4. Simplify the Expression:
Thus, the expression simplifies to:
[tex]\[
a^{\frac{2}{15}}
\][/tex]
The simplest form of the expression [tex]\(\frac{\sqrt[5]{a^4}}{\sqrt[3]{a^2}}\)[/tex] is:
[tex]\[
\boxed{a^{\frac{2}{15}}}
\][/tex]
