# Find the following products and express answers in simplest radical form. All va

Find the following products and express answers in simplest radical form. All variables represent nonnegative real numbers.
$$\displaystyle{\left({3}\sqrt{{{2}}}-{5}\sqrt{{{3}}}\right)}{\left({6}\sqrt{{{2}}}-{7}\sqrt{{{3}}}\right)}$$

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Give that,
$$\displaystyle{\left({3}\sqrt{{{2}}}-{5}\sqrt{{{3}}}\right)}{\left({6}\sqrt{{{2}}}-{7}\sqrt{{{3}}}\right)}$$
Consider,
$$\displaystyle{\left({3}\sqrt{{{2}}}-{5}\sqrt{{{3}}}\right)}{\left({6}\sqrt{{{2}}}-{7}\sqrt{{{3}}}\right)}={3}\sqrt{{{2}}}{\left({6}\sqrt{{{2}}}-{7}\sqrt{{{3}}}\right)}-{5}\sqrt{{{3}}}{\left({6}\sqrt{{{2}}}-{7}\sqrt{{{3}}}\right)}$$
$$\displaystyle={3}\sqrt{{{2}}}{\left({6}\sqrt{{{2}}}\right)}-{3}\sqrt{{{2}}}{\left({7}\sqrt{{{3}}}\right)}-{5}\sqrt{{{3}}}{\left({6}\sqrt{{{2}}}\right)}+{5}\sqrt{{{3}}}{\left({7}\sqrt{{{3}}}\right)}$$
$$\displaystyle={18}\times{2}-{21}\sqrt{{{6}}}-{30}\sqrt{{{6}}}+{35}\times{3}$$
$$\displaystyle={36}+{105}-\sqrt{{{6}}}{\left({21}+{30}\right)}$$
$$\displaystyle={141}-{51}\sqrt{{{6}}}$$
Answer: $$\displaystyle{\left({3}\sqrt{{{2}}}-{5}\sqrt{{{3}}}\right)}{\left({6}\sqrt{{{2}}}-{7}\sqrt{{{3}}}\right)}={141}-{51}\sqrt{{{6}}}$$