What is the following product?
$(2 \sqrt{7}+3 \sqrt{6})(5 \sqrt{2}+4 \sqrt{3})$
A. $6 \sqrt{10}+16 \sqrt{2}+42$
B. $8 \sqrt{10}+30 \sqrt{2}+66$
C. $7 \sqrt{14}+6 \sqrt{21}+16 \sqrt{3}+21 \sqrt{2}$
D. $10 \sqrt{14}+8 \sqrt{21}+30 \sqrt{3}+36 \sqrt{2}$
Answer (2)
The product of ( 2 7 + 3 6 ) ( 5 2 + 4 3 ) is found by distributing each term, resulting in 10 14 + 8 21 + 30 3 + 36 2 , which corresponds to option D. Therefore, the correct answer is option D.
;
Expand the product using the distributive property (FOIL method).
Simplify each term in the expansion.
Combine the simplified terms.
The final answer is 10 14 + 8 21 + 30 3 + 36 2 .
Explanation
Understanding the Problem We are given the product of two expressions: $(2
\sqrt{7}+3 \sqrt{6})(5 \sqrt{2}+4 \sqrt{3})$. Our goal is to expand this product and simplify it to match one of the given options. We will use the distributive property (also known as the FOIL method) to expand the product.
Expanding the Product Let's expand the product ( 2 7 + 3 6 ) ( 5 2 + 4 3 ) :
First, multiply each term in the first parentheses by each term in the second parentheses:
( 2 7 ) ( 5 2 ) + ( 2 7 ) ( 4 3 ) + ( 3 6 ) ( 5 2 ) + ( 3 6 ) ( 4 3 )
Simplifying Each Term Now, let's simplify each term:
( 2 7 ) ( 5 2 ) = 10 14
( 2 7 ) ( 4 3 ) = 8 21
( 3 6 ) ( 5 2 ) = 15 12 = 15 4 × 3 = 15 × 2 3 = 30 3
( 3 6 ) ( 4 3 ) = 12 18 = 12 9 × 2 = 12 × 3 2 = 36 2
Combining Like Terms Now, combine the simplified terms:
10 14 + 8 21 + 30 3 + 36 2
Finding the Correct Answer Comparing our simplified expression 10 14 + 8 21 + 30 3 + 36 2 with the given options, we find that it matches the last option:
10 14 + 8 21 + 30 3 + 36 2
Final Answer Therefore, the correct answer is 10 14 + 8 21 + 30 3 + 36 2 .
Examples
This type of problem, involving the multiplication of expressions with square roots, is useful in various fields such as physics and engineering when dealing with distances, areas, or volumes in geometric calculations. For example, when calculating the area of a rectangle with sides of length ( 2 7 + 3 6 ) and ( 5 2 + 4 3 ) , you would need to perform this multiplication. Similarly, in physics, you might encounter such expressions when calculating energy levels or wave functions.
