Q. 8.76 - Intermediate Algebra

Question

Simplify

1. $\sqrt{27}+4\sqrt{3}$
   
2. $4\sqrt[3]{5}-7\sqrt[3]{40}$ 
3. $\frac{1}{2}\sqrt[3]{128}-\frac{5}{3}\sqrt[3]{54}$
   

Step-by-Step Solution

Answer

Part a. $7\sqrt{3}$

Part b. $-10\sqrt[3]{5}$

Part c. $-3\sqrt[3]{2}$

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

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

$\begin{array}{rcl}\sqrt{27}+4\sqrt{3}& =& \sqrt{9·3}+4\sqrt{3}\\ & =& \sqrt{9}·\sqrt{3}+4\sqrt{3}\\ & =& 3\sqrt{3}+4\sqrt{3}\end{array}$

On combing the like radicals we get

$\begin{array}{rcl}3\sqrt{3}+4\sqrt{3}& =& (3+4)\sqrt{3}\\ & =& 7\sqrt{3}\end{array}$

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

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

$\begin{array}{rcl}4\sqrt[3]{5}-7\sqrt[3]{40}& =& 4\sqrt[3]{5}-7\sqrt[3]{8·5}\\ & =& 4\sqrt[3]{5}-7\sqrt[3]{8}·\sqrt[3]{5}\\ & =& 4\sqrt[3]{5}-7·2·\sqrt[3]{5}\\ & =& 4\sqrt[3]{5}-14\sqrt[3]{5}\end{array}$

On combing the like radicals we get

$\begin{array}{rcl}4\sqrt[3]{5}-14\sqrt[3]{5}& =& (4-14)\sqrt[3]{5}\\ & =& -10\sqrt[3]{5}\end{array}$

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

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

$\begin{array}{rcl}\frac{1}{2}\sqrt[3]{128}-\frac{5}{3}\sqrt[3]{54}& =& \frac{1}{2}\sqrt[3]{128}-\frac{5}{3}\sqrt[3]{54}\\ & =& \frac{1}{2}\sqrt[3]{64·2}-\frac{5}{3}\sqrt[3]{27·2}\\ & =& \frac{1}{2}\sqrt[3]{64}·\sqrt[3]{2}-\frac{5}{3}\sqrt[3]{27}·\sqrt[3]{2}\\ & =& \frac{1}{2}·4·\sqrt[3]{2}-\frac{5}{3}·3·\sqrt[3]{2}\\ & =& 2\sqrt[3]{2}-5\sqrt[3]{2}\end{array}$

On combing the like radicals we get

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