Rewrite the root expressions as fractional exponents:
[tex]\dfrac{\sqrt[3]{7}}{\sqrt[5]{7}} = \dfrac{7^{1/3}}{7^{1/5}}[/tex]
Recall that [tex]\frac{a^m}{a^n} = a^{m-n}[/tex], so that
[tex]\dfrac{7^{1/3}}{7^{1/5}} = 7^{1/3 - 1/5}[/tex]
Simplify the exponent:
[tex]\dfrac13 - \dfrac15 = \dfrac5{15} - \dfrac3{15} = \dfrac{5-3}{15} = \dfrac2{15}[/tex]
Then you end up with
[tex]\dfrac{\sqrt[3]{7}}{\sqrt[5]{7}} = 7^{2/15}[/tex]
