Q 8.75 - Intermediate Algebra

Question

Simplify:

1. $\sqrt{18}+6\sqrt{2}$
   
2. $6\sqrt[3]{16}-2\sqrt[3]{250}$
   
3. $\frac{2}{3}\sqrt[3]{81}-\frac{1}{2}\sqrt[3]{24}$
   

Step-by-Step Solution

Answer

Part a. $9\sqrt{2}$

Part b. $2\sqrt[3]{2}$

Part c. $\sqrt[3]{3}$ 

Part (a) Step 1. Simplify the radicals and combine like radicals

Using $\sqrt{ab}=\sqrt{a}·\sqrt{b}$ simplify the expression.

$\begin{array}{rcl}\sqrt{18}+6\sqrt{2}& =& \sqrt{9·2}+6\sqrt{2}\\ & =& \sqrt{9}·\sqrt{2}+6\sqrt{2}\\ & =& 3\sqrt{2}+6\sqrt{2}\end{array}$

On combing the like radicals we get

$\begin{array}{rcl}3\sqrt{2}+6\sqrt{2}& =& (3+6)\sqrt{2}\\ & =& 9\sqrt{2}\end{array}$

Part (b) Step 1. Simplify the radicals and combine like radicals

Using $\sqrt{ab}=\sqrt{a}·\sqrt{b}$ simplify the expression.

$\begin{array}{rcl}6\sqrt[3]{16}-2\sqrt[3]{250}& =& 6\sqrt[3]{8·2}-2\sqrt[3]{125·2}\\ & =& 6\sqrt[3]{8}·\sqrt[3]{2}-2\sqrt[3]{125}·\sqrt[3]{2}\\ & =& 6·2·\sqrt[3]{2}-2·5·\sqrt[3]{2}\\ & =& 12\sqrt[3]{2}-10\sqrt[3]{2}\end{array}$

On combing the like radicals we get

$\begin{array}{rcl}12\sqrt[3]{2}-10\sqrt[3]{2}& =& (12-10)\sqrt[3]{2}\\ & =& 2\sqrt[3]{2}\end{array}$

Part (c) Step 1. Simplify the radicals and combine like radicals

Using $\sqrt{ab}=\sqrt{a}·\sqrt{b}$ simplify the expression.

$\begin{array}{rcl}\frac{2}{3}\sqrt[3]{81}-\frac{1}{2}\sqrt[3]{24}& =& \frac{2}{3}\sqrt[3]{27·3}-\frac{1}{2}\sqrt[3]{8·3}\\ & =& \frac{2}{3}\sqrt[3]{27}·\sqrt[3]{3}-\frac{1}{2}\sqrt[3]{8}·\sqrt[3]{3}\\ & =& \frac{2}{3}·3·\sqrt[3]{3}-\frac{1}{2}·2·\sqrt[3]{3}\\ & =& 2\sqrt[3]{3}-\sqrt[3]{3}\end{array}$

On combing the like radicals we get

$\begin{array}{rcl}2\sqrt[3]{3}-\sqrt[3]{3}& =& (2-1)\sqrt[3]{3}\\ & =& \sqrt[3]{3}\end{array}$