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.
A)$\frac { \sqrt [ 6 ] { 4 } } { \sqrt { 7 } }$
B) $\frac { \sqrt [ 7 ] { 1372 } } { 7 }$
C) $\frac { \sqrt [ 6 ] { 1372 } } { 4 }$
D) $ \frac { \sqrt [ 6 ] { 1372 } } { 7 }$

Question

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. 
A)$\frac { \sqrt [ 6 ] { 4 } } { \sqrt { 7 } }$
B) $\frac { \sqrt [ 7 ] { 1372 } } { 7 }$
C) $\frac { \sqrt [ 6 ] { 1372 } } { 4 }$
D) $ \frac { \sqrt [ 6 ] { 1372 } } { 7 }$

Quizplus · Accepted Answer

To combine the radicals, we first express them with a common index. The expression is $\frac{\sqrt[6]{4}}{\sqrt{7}}$. The square root of 7 can be written as $\sqrt[6]{7^3}$ because raising 7 to the 3rd power and taking the 6th root is equivalent to taking the square root. Thus, the expression becomes $\frac{\sqrt[6]{4}}{\sqrt[6]{7^3}} = \sqrt[6]{\frac{4}{7^3}}$. Simplifying inside the radical gives $\sqrt[6]{\frac{4}{343}}$. Multiplying the numerator and denominator inside the radical by 4 gives $\sqrt[6]{\frac{16}{1372}}$, which simplifies to $\frac{\sqrt[6]{1372}}{7}$, matching option D.

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