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. $\frac { \sqrt [ 4 ] { 2 } } { \sqrt { 6 } }$
A)$\frac { \sqrt [ 4 ] { 72 } } { 6 }$
B) $\frac { \sqrt [ 5 ] { 72 } } { 6 }$
C) $\frac { \sqrt [ 4 ] { 72 } } { 8 }$
D) $\frac { \sqrt [ 4 ] { 72 } } { 2 }$

Question

Quizplus · Accepted Answer

To write the expression as a single radical, we first express both radicals with a common index. The given expression is $\frac { \sqrt [ 4 ] { 2 } } { \sqrt { 6 } }$. The square root can be written as a fourth root by raising the radicand to the power of 2: $\sqrt{6} = \sqrt[4]{6^2} = \sqrt[4]{36}$. Thus, the expression becomes $\frac { \sqrt [ 4 ] { 2 } } { \sqrt [ 4 ] { 36 } }$, which simplifies to $\sqrt[4]{\frac{2}{36}} = \sqrt[4]{\frac{1}{18}}$. To get rid of the fraction inside the radical, we multiply both numerator and denominator by 18, leading to $\sqrt[4]{\frac{18}{18^2}} = \sqrt[4]{\frac{18}{324}}$. Since 18 is 2*9 and 324 is 36*9, and 36 is 6 squared, we simplify this to $\sqrt[4]{\frac{2*9}{36*9}} = \sqrt[4]{\frac{2}{36}}$, which was a simplification step to illustrate the process. However, the correct approach directly from the original expression should be to multiply the numerator and denominator by $\sqrt[4]{36}$ to get a common denominator, leading to $\frac{\sqrt[4]{2}*\sqrt[4]{36}}{\sqrt[4]{36}*\sqrt[4]{36}} = \frac{\sqrt[4]{72}}{\sqrt[4]{1296}}$. Since $1296 = 6^4$, the denominator simplifies to 6, resulting in $\frac{\sqrt[4]{72}}{6}$, which matches choice A. The initial explanation contained a misstep in simplification; the correct process involves recognizing that multiplying the numerator and denominator by the appropriate power of the radicand to achieve a common index and then simplifying directly leads to the answer without the need for intermediate fraction simplification as initially described.

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