Chapter 9 - Roots and Radicals - 9.4 - Products and Quotients Involving Radicals - Problem Set 9.4 - Page 418: 34

$1-5\sqrt[]{7}$

Using the FOIL method and the properties of radicals, the given expression, $ (\sqrt[]{7}-6)(\sqrt[]{7}+1) ,$ simplifies to \begin{array}{l}\require{cancel} \sqrt[]{7}(\sqrt[]{7})+\sqrt[]{7}(1)-6(\sqrt[]{7})-6(1) \\\\= 7+\sqrt[]{7}-6\sqrt[]{7}-6 \\\\= (7-6)+(\sqrt[]{7}-6\sqrt[]{7}) \\\\= 1-5\sqrt[]{7} .\end{array}