Question

Simplify cube root of 7 over fifth root of 7.

Answer 1

So to put your equation into algebraic terms, your asking for $\frac{\sqrt[3]{7}}{\sqrt[5]{7}}$ .

Firstly, we have to convert these into fractional exponents. The rule to do that is $x^{\frac{m}{n}}=\sqrt[n]{x^m}$ . Applying that here, our equation is $\frac{7^{\frac{1}{3}}}{7^{\frac{1}{5}}}$

Next, the rule with dividing exponents with the same base is to just subtract the exponents, so with this we are subtracting 1/5 from 1/3. However, we need to find their LCM, or lowest common multiple, of 3 and 5. You can do this by listing out what numbers 3 and 5 are factors of. In this case, the LCM is 15. Multiply 1/3 by 5/5 and 1/5 by 3/3:

$\frac{1}{3}*\frac{5}{5}=\frac{5}{15}\\ \\ \frac{1}{5}*\frac{3}{3}=\frac{3}{15}\\ \\ \frac{7^{\frac{5}{15}}}{7^{\frac{3}{15}}}$

Now that they share the same denominator, subtract the numerators of the 2 fractional exponents and your answer will be $7^\frac{2}{15}$, or the last option.