Question

Simplify:

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

Short answer

Part a.$18+\sqrt{6}$

Part b.$\begin{array}{rcl}& & -2\sqrt[3]{4}-2\sqrt[3]{3}\end{array}$

Step-by-step solution

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

First, expand the expression using the distributive property.

$\begin{array}{rcl}\sqrt{6}(1+3\sqrt{6})& =& \sqrt{6}·1+\sqrt{6}·3\sqrt{6}\\ & =& \sqrt{6}+3·{\left(\sqrt{6}\right)}^2=\sqrt{6}+3·6\\ & =& \sqrt{6}+18\end{array}$

There are no like radicals, so the expression cannot be simplified further.

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

First, expand the expression using the distributive property.

$\begin{array}{rcl}\sqrt[3]{4}(-2-\sqrt[3]{6})& =& \sqrt[3]{4}·(-2)-\sqrt[3]{4}·\sqrt[3]{6}\\ & =& -2\sqrt[3]{4}-\sqrt[3]{4·6}\\ & =& -2\sqrt[3]{4}-\sqrt[3]{24}\end{array}$

Simplify the second term using the product property of radical as

$\begin{array}{rcl}-2\sqrt[3]{4}-\sqrt[3]{24}& =& -2\sqrt[3]{4}-\sqrt[3]{8·3}\\ & =& -2\sqrt[3]{4}-\sqrt[3]{8}·\sqrt[3]{3}\\ & =& -2\sqrt[3]{4}-2\sqrt[3]{3}\end{array}$