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What is the following product?

[tex]\[
(2 \sqrt{7} + 3 \sqrt{6})(5 \sqrt{2} + 4 \sqrt{3})
\][/tex]

A. [tex]\(6 \sqrt{10} + 16 \sqrt{2} + 42\)[/tex]

B. [tex]\(8 \sqrt{10} + 30 \sqrt{2} + 66\)[/tex]

C. [tex]\(7 \sqrt{14} + 6 \sqrt{21} + 16 \sqrt{3} + 21 \sqrt{2}\)[/tex]

D. [tex]\(10 \sqrt{14} + 8 \sqrt{21} + 30 \sqrt{3} + 36 \sqrt{2}\)[/tex]

Sagot :

To find the product of the expression [tex]\((2\sqrt{7} + 3\sqrt{6})(5\sqrt{2} + 4\sqrt{3})\)[/tex], follow these detailed steps:

1. Start by expanding the product using the distributive property (FOIL method for binomials):
[tex]\[ (2\sqrt{7} + 3\sqrt{6})(5\sqrt{2} + 4\sqrt{3}) = 2\sqrt{7} \cdot 5\sqrt{2} + 2\sqrt{7} \cdot 4\sqrt{3} + 3\sqrt{6} \cdot 5\sqrt{2} + 3\sqrt{6} \cdot 4\sqrt{3} \][/tex]

2. Calculate each multiplication individually:
- For the first term:
[tex]\[ 2\sqrt{7} \cdot 5\sqrt{2} = 2 \cdot 5 \cdot \sqrt{7 \cdot 2} = 10\sqrt{14} \][/tex]

- For the second term:
[tex]\[ 2\sqrt{7} \cdot 4\sqrt{3} = 2 \cdot 4 \cdot \sqrt{7 \cdot 3} = 8\sqrt{21} \][/tex]

- For the third term:
[tex]\[ 3\sqrt{6} \cdot 5\sqrt{2} = 3 \cdot 5 \cdot \sqrt{6 \cdot 2} = 15\sqrt{12} \][/tex]
Simplify [tex]\(\sqrt{12}\)[/tex] to get:
[tex]\[ 15\sqrt{12} = 15 \cdot 2\sqrt{3} = 30\sqrt{3} \][/tex]

- For the fourth term:
[tex]\[ 3\sqrt{6} \cdot 4\sqrt{3} = 3 \cdot 4 \cdot \sqrt{6 \cdot 3} = 12\sqrt{18} \][/tex]
Simplify [tex]\(\sqrt{18}\)[/tex] to get:
[tex]\[ 12\sqrt{18} = 12 \cdot 3\sqrt{2} = 36\sqrt{2} \][/tex]

3. Combine all terms to form the expanded expression:
[tex]\[ (2\sqrt{7} + 3\sqrt{6})(5\sqrt{2} + 4\sqrt{3}) = 10\sqrt{14} + 8\sqrt{21} + 30\sqrt{3} + 36\sqrt{2} \][/tex]

4. The expanded product of the given expression is:
[tex]\[ 10\sqrt{14} + 8\sqrt{21} + 30\sqrt{3} + 36\sqrt{2} \][/tex]

This matches the fourth provided option:
[tex]\[ 10 \sqrt{14} + 8 \sqrt{21} + 30 \sqrt{3} + 36 \sqrt{2} \][/tex]

Thus, the correct answer is:
[tex]\[ 10 \sqrt{14} + 8 \sqrt{21} + 30 \sqrt{3} + 36 \sqrt{2} \][/tex]