For this case we must find the product of the following expression:
[tex]\sqrt [3] {5} * \sqrt {2}[/tex]
By definition of properties of powers and roots we have:
[tex]\sqrt [n] {a ^ m} = a ^ {\frac {m} {n}}[/tex]
We rewrite the expression using the lowest common index of 6, then:
[tex]5 ^ {\frac {1} {3}} * 2 ^ {\frac {1} {2}} =[/tex]
We rewrite the terms in an equivalent way:
[tex]5 ^ {\frac {2} {6}} * 2 ^ {\frac {3} {6}} =[/tex]
We rewrite the expression using the property mentioned:
[tex]\sqrt [6] {5 ^ 2} * \sqrt [6] {2 ^ 3} =[/tex]
We combine using the product rule for radicals:
[tex]\sqrt [n] {a} * \sqrt [n] {b} = \sqrt [n] {ab}[/tex]
So:
[tex]\sqrt [6] {5 ^ 2 * 2 ^ 3} =\\\sqrt [6] {25 * 8} =\\\sqrt[6]{200}[/tex]
ANswer:
Option b
