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]
