Question

Simplify:

1. $(6-3\sqrt{7})(3+4\sqrt{7})$
2. $(\sqrt[3]{x}-2)(\sqrt[3]{x}-3)$
   

Short answer

Part a.$-66+15\sqrt{7}$

Part b.$\begin{array}{rcl}& & \sqrt[3]{x^2}-5\sqrt[3]{x}+6\end{array}$

Step-by-step solution

Part (a) Step 1. Expand the expression and simplify the radicals 

Multiply and expand the expression.

$\begin{array}{rcl}(6-3\sqrt{7})(3+4\sqrt{7})& =& 18+24\sqrt{7}-9\sqrt{7}-12{\left(\sqrt{7}\right)}^2\\ & =& 18+24\sqrt{7}-9\sqrt{7}-84\end{array}$

Now combine the like numbers and like radicals to get the simplified answer.

$\begin{array}{rcl}18+24\sqrt{7}-9\sqrt{7}-84& =& (18-84)+(24\sqrt{7}-9\sqrt{7})\\ & =& -66+(24-9)\sqrt{7}\\ & =& -66+15\sqrt{7}\end{array}$

Part (b) Step 1. Expand the expression and simplify the radicals 

Multiply and expand the expression.

$\begin{array}{rcl}(\sqrt[3]{x}-2)(\sqrt[3]{x}-3)& =& \sqrt[3]{x}·\sqrt[3]{x}-3\sqrt[3]{x}-2\sqrt[3]{x}+6\\ & =& \sqrt[3]{x^2}-3\sqrt[3]{x}-2\sqrt[3]{x}+6\end{array}$

Now combine the like radicals to get the simplifid expression.

$\begin{array}{rcl}\sqrt[3]{x^2}-3\sqrt[3]{x}-2\sqrt[3]{x}+6& =& \sqrt[3]{x^2}-(3+2)\sqrt[3]{x}+6\\ & =& \sqrt[3]{x^2}-5\sqrt[3]{x}+6\end{array}$