Q. 8.80 - Intermediate Algebra

Question

Simplify:

1. $\left(3\sqrt{3}\right)\left(3\sqrt{6}\right)$
   
2. $(-4\sqrt[3]{9})\left(3\sqrt[3]{6}\right)$
   

Step-by-Step Solution

Answer

Part a. $27\sqrt{2}$

Part b. $\begin{array}{rcl}& & -36\sqrt[3]{2}\end{array}$

Part (a) Step 1. Multiply the radicals and simplify

Using the product property of roots that is $\sqrt[n]{a}·\sqrt[n]{b}=\sqrt[n]{ab}$, the expression can be simplified to

$\begin{array}{rcl}\left(3\sqrt{3}\right)\left(3\sqrt{6}\right)& =& (3·3)(\sqrt{3}·\sqrt{6})\\ & =& 9\sqrt{3·6}\\ & =& 9\sqrt{18}\end{array}$

Now, the radical can be simplified as

$\begin{array}{rcl}9\sqrt{18}& =& 9\sqrt{9·2}\\ & =& 9·\sqrt{9}·\sqrt{2}\\ & =& 9·3\sqrt{2}\\ & =& 27\sqrt{2}\end{array}$

Part (b) Step 1. Multiply the radicals and simplify

Using the product property of roots that is $\sqrt[n]{a}·\sqrt[n]{b}=\sqrt[n]{ab}$, the expression can be simplified to

$\begin{array}{rcl}(-4\sqrt[3]{9})\left(3\sqrt[3]{6}\right)& =& (-4·3)(\sqrt[3]{9}·\sqrt[3]{6})\\ & =& -12\sqrt[3]{9·6}\\ & =& -12\sqrt[3]{54}\end{array}$

Now, the radical can be simplified as

$\begin{array}{rcl}-12\sqrt[3]{54}& =& -12\sqrt[3]{27·2}\\ & =& -12\sqrt[3]{27}·\sqrt[3]{2}\\ & =& -12·3·\sqrt[3]{2}\\ & =& -36\sqrt[3]{2}\end{array}$