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.
$$\sqrt { 4 } \sqrt [ 3 ] { 4 }$$
A) $\sqrt { 1024 }$
B) $\sqrt [ 6 ] { 1024 }$
C) $\sqrt [ 3 ] { 1024 }$
D) $\sqrt [ 6 ] { 4 }$
E) $\sqrt [ 6 ] { 256 }$

Question

Quizplus · Accepted Answer

We can write $\sqrt{4}$ as $\sqrt[6]{16}$ and $\sqrt[3]{4}$ as $\sqrt[6]{8}$. Then, using the product of radicals property, we have $\sqrt{4}\sqrt[3]{4}=\sqrt[6]{16}\sqrt[6]{8}=\sqrt[6]{128}= \sqrt[6]{1024}$. Therefore, the answer is choice B.

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 $\sqrt { 3 } \sqrt [ 3 ] { 5 } = \sqrt [ 6 ] { 27 } \sqrt [ 6 ] { 25 } = \sqrt [ 6 ] { ( 27 ) ( 25 ) } = \sqrt [ 6 ] { 675 }$