To simplify the expression [tex]\(\frac{\sqrt[3]{7}}{\sqrt[5]{7}}\)[/tex], let's proceed step by step.
1. Convert the roots to exponents:
- [tex]\(\sqrt[3]{7}\)[/tex] can be written as [tex]\(7^{\frac{1}{3}}\)[/tex]
- [tex]\(\sqrt[5]{7}\)[/tex] can be written as [tex]\(7^{\frac{1}{5}}\)[/tex]
2. Express the fraction using exponents:
[tex]\[\frac{7^{\frac{1}{3}}}{7^{\frac{1}{5}}}\][/tex]
3. Apply the properties of exponents:
When you divide two numbers with the same base, you subtract the exponents:
[tex]\[7^{\frac{1}{3} - \frac{1}{5}}\][/tex]
4. Calculate the exponent:
- Find a common denominator for [tex]\(\frac{1}{3}\)[/tex] and [tex]\(\frac{1}{5}\)[/tex]. The least common multiple of 3 and 5 is 15.
- Rewrite the fractions with a common denominator:
[tex]\[\frac{1}{3} = \frac{5}{15}\][/tex]
[tex]\[\frac{1}{5} = \frac{3}{15}\][/tex]
- Subtract the exponents:
[tex]\[\frac{5}{15} - \frac{3}{15} = \frac{2}{15}\][/tex]
5. Write the result:
[tex]\[\frac{\sqrt[3]{7}}{\sqrt[5]{7}} = 7^{\frac{2}{15}}\][/tex]
So, the simplified form of [tex]\(\frac{\sqrt[3]{7}}{\sqrt[5]{7}}\)[/tex] is [tex]\(7^{\frac{2}{15}}\)[/tex].
