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Which of the following is the simplest form of this expression? [tex]\frac{\sqrt[5]{a^4}}{\sqrt[3]{a^2}}[/tex]

A. [tex]a^{\frac{2}{15}}[/tex]

B. [tex]\frac{1}{a^{\frac{7}{4}}}[/tex]

C. [tex]a^{\frac{7}{4}}[/tex]

D. [tex]\frac{1}{a^{\frac{2}{15}}}[/tex]


Sagot :

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]