We can often multiply and divide radicals with different indexes.For example: $$\sqrt { 3 } \sqrt [ 3 ] { 5 } = \sqrt [ 6 ] { 27 } \sqrt [ 6 ] { 25 } = \sqrt [ 6 ] { ( 27 ) ( 25 ) } = \sqrt [ 6 ] { 675 }$$
Use this idea to write the following expression as a single radical.
$\sqrt { 5 } \sqrt [ 3 ] { 5 }$
A) $\sqrt [ 3 ] { 3125 }$
B) $\sqrt [ 6 ] { 5 }$
C) $\sqrt [ 6 ] { 3125 }$
D) $\sqrt { 3125 }$
E) $\sqrt [ 6 ] { 625 }$

Question

Quizplus · Accepted Answer

We can write $\sqrt{5}$ as $\sqrt[6]{5^3}$ and $\sqrt[3]{5}$ as $\sqrt[6]{5^2}$. Therefore, $\sqrt{5}\sqrt[3]{5}=\sqrt[6]{5^3}\sqrt[6]{5^2}=\sqrt[6]{5^5}=\sqrt[6]{3125}$, which corresponds to choice $\boxed{	extbf{(C)}}$.

We Can Often Multiply and Divide Radicals with Different Indexes 353=276256=(27)(25)6=6756\sqrt { 3 } \sqrt [ 3 ] { 5 } = \sqrt [ 6 ] { 27 } \sqrt [ 6 ] { 25 } = \sqrt [ 6 ] { ( 27 ) ( 25 ) } = \sqrt [ 6 ] { 675 }3​35​=627​625​=6(27)(25)​=6675​

We Can Often Multiply and Divide Radicals with Different Indexes $\sqrt { 3 } \sqrt [ 3 ] { 5 } = \sqrt [ 6 ] { 27 } \sqrt [ 6 ] { 25 } = \sqrt [ 6 ] { ( 27 ) ( 25 ) } = \sqrt [ 6 ] { 675 }$