Chapter 9 - Roots and Radicals - 9.4 - Products and Quotients Involving Radicals - Problem Set 9.4 - Page 418: 42

$-95$

Using $(a+b)(a-b)=a^2-b^2$, or the special product of the sum and difference of like terms, the given expression, $ (4\sqrt{5}+5\sqrt{7})(4\sqrt{5}-5\sqrt{7}) ,$ simplifies to \begin{array}{l}\require{cancel} (4\sqrt{5})^2-(5\sqrt{7})^2 \\\\= 16(5)-25(7) \\\\= 80-175 \\\\= -95 .\end{array}