Find the following product and express your answer in simplest radical form. $( 3 \sqrt { 7 } - 4 \sqrt { 21 } ) ( 5 \sqrt { 27 } + \sqrt { 21 } )$
A) $- 45 \sqrt { 21 } - 21 \sqrt { 3 } + 180 \sqrt { 7 } + 84$
B) $45 \sqrt { 21 } - 21 \sqrt { 7 } - 180 \sqrt { 3 } + 84$
C) $15 \sqrt { 3 } + 3 \sqrt { 7 } - 84$
D) $45 \sqrt { 21 } + 21 \sqrt { 3 } - 180 \sqrt { 7 } - 84 \sqrt { 27 }$
E) $45 \sqrt { 21 } + 21 \sqrt { 3 } - 180 \sqrt { 7 } - 84$

Question

Quizplus · Accepted Answer

First, simplify the radicals where possible: $5\sqrt{27} = 5\sqrt{9 \cdot 3} = 15\sqrt{3}$ and $4\sqrt{21}$ remains as is because it cannot be simplified further. Then, multiply the terms using the distributive property: $3\sqrt{7} \cdot 15\sqrt{3} = 45\sqrt{21}$, $3\sqrt{7} \cdot \sqrt{21} = 3\sqrt{147} = 21\sqrt{3}$, $-4\sqrt{21} \cdot 15\sqrt{3} = -60\sqrt{63} = -180\sqrt{7}$, and $-4\sqrt{21} \cdot \sqrt{21} = -4 \cdot 21 = -84$. Adding these results gives $45\sqrt{21} + 21\sqrt{3} - 180\sqrt{7} - 84$, which matches option E.

Find the Following Product and Express Your Answer in Simplest (37−421)(527+21)( 3 \sqrt { 7 } - 4 \sqrt { 21 } ) ( 5 \sqrt { 27 } + \sqrt { 21 } )(37​−421​)(527​+21​)

Find the Following Product and Express Your Answer in Simplest $( 3 \sqrt { 7 } - 4 \sqrt { 21 } ) ( 5 \sqrt { 27 } + \sqrt { 21 } )$