Question

Please help thanks!!

Answer 1

Think kind of problems can be solved by turning roots into rational exponents. If you remember that

$\sqrt[n]{a^m} = a^{\frac{m}{n}}$, you have

$\sqrt{5} = 5^{\frac{1}{2}}, \sqrt[3]{5} = 5^{\frac{1}{3}}$

Moreover, recall the following rule for multiplying powers of the same base:

$a^b \cdot a^c = a^{b+c}$

to write

$\sqrt{5}\cdot \sqrt[3]{5} = 5^{\frac{1}{2}} \cdot 5^{\frac{1}{3}} = 5^{\frac{1}{2}+\frac{1}{3}}$

since $\cfrac{1}{2}+\cfrac{1}{3} = \cfrac{5}{6}$, the result is

$5^{\frac{5}{6}} = \sqrt[6]{5^5}$