To simplify the expression [tex]\(\frac{\sqrt[5]{a^4}}{\sqrt[3]{a^2}}\)[/tex], let's break down each part step-by-step.
1. Rewrite the roots as exponents:
- The fifth root of [tex]\(a^4\)[/tex] can be written as [tex]\(a^{4/5}\)[/tex].
- The cube root of [tex]\(a^2\)[/tex] can be written as [tex]\(a^{2/3}\)[/tex].
So, the expression becomes:
[tex]\[
\frac{a^{4/5}}{a^{2/3}}
\][/tex]
2. Simplify the fraction:
When you divide two expressions with the same base, you subtract the exponents:
[tex]\[
a^{\frac{4}{5}} \div a^{\frac{2}{3}} = a^{\frac{4}{5} - \frac{2}{3}}
\][/tex]
3. Subtract the exponents:
To subtract [tex]\(\frac{4}{5}\)[/tex] and [tex]\(\frac{2}{3}\)[/tex], we need a common denominator. The least common denominator of 5 and 3 is 15.
- Convert [tex]\(\frac{4}{5}\)[/tex] to have a denominator of 15:
[tex]\[
\frac{4}{5} = \frac{4 \times 3}{5 \times 3} = \frac{12}{15}
\][/tex]
- Convert [tex]\(\frac{2}{3}\)[/tex] to have a denominator of 15:
[tex]\[
\frac{2}{3} = \frac{2 \times 5}{3 \times 5} = \frac{10}{15}
\][/tex]
- Subtract the fractions:
[tex]\[
\frac{12}{15} - \frac{10}{15} = \frac{2}{15}
\][/tex]
4. Write the simplified expression:
The simplified exponent is [tex]\(\frac{2}{15}\)[/tex], so the expression simplifies to:
[tex]\[
a^{\frac{2}{15}}
\][/tex]
Therefore, the simplest form of [tex]\(\frac{\sqrt[5]{a^4}}{\sqrt[3]{a^2}}\)[/tex] is:
Answer:
[tex]\[
\boxed{a^{\frac{2}{15}}}
\][/tex]
Thus, the correct answer is option A. [tex]\(a^{\frac{2}{15}}\)[/tex].
