What is the following product?

[tex]\[ (5 \sqrt{2} - 4 \sqrt{3})(5 \sqrt{2} - 4 \sqrt{3}) \][/tex]

A. \(-2\)

B. \(2\)

C. \(22 - 40 \sqrt{6}\)

D. [tex]\(98 - 40 \sqrt{6}\)[/tex]



Answer :

To find the product \((5 \sqrt{2} - 4 \sqrt{3})(5 \sqrt{2} - 4 \sqrt{3})\), we need to expand the expression using the distributive property (also known as the FOIL method for binomials):

[tex]\[ (5 \sqrt{2} - 4 \sqrt{3})(5 \sqrt{2} - 4 \sqrt{3}) \][/tex]

Step-by-step solution for expansion:

1. First: Multiply the first terms in each binomial:
[tex]\[ (5 \sqrt{2})(5 \sqrt{2}) = 25 \cdot 2 = 50 \][/tex]

2. Outer: Multiply the outer terms in the binomials:
[tex]\[ (5 \sqrt{2})(-4 \sqrt{3}) = -20 \sqrt{6} \][/tex]

3. Inner: Multiply the inner terms in the binomials:
[tex]\[ (-4 \sqrt{3})(5 \sqrt{2}) = -20 \sqrt{6} \][/tex]

4. Last: Multiply the last terms in each binomial:
[tex]\[ (-4 \sqrt{3})(-4 \sqrt{3}) = 16 \cdot 3 = 48 \][/tex]

Now we combine all these results together:
[tex]\[ 50 + (-20 \sqrt{6}) + (-20 \sqrt{6}) + 48 \][/tex]

Combine like terms:
[tex]\[ 50 + 48 - 40 \sqrt{6} \][/tex]

Sum the constant terms:
[tex]\[ 98 - 40 \sqrt{6} \][/tex]

So, the final answer for the product \((5 \sqrt{2} - 4 \sqrt{3})(5 \sqrt{2} - 4 \sqrt{3})\) is:
[tex]\[ 98 - 40 \sqrt{6} \][/tex]

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