Question

Simplify:

1. $\sqrt{32m^7}-\sqrt{50m^7}$
2. $\sqrt[3]{135x^7}-\sqrt[3]{40x^7}$
   

Short answer

Part a.$-m^3\sqrt{2m}$

Part b. localid="1645681976813" $x^2\sqrt[3]{5x}$

Step-by-step solution

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

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

$\begin{array}{rcl}\sqrt{32m^7}-\sqrt{50m^7}& =& \sqrt{16m^6·2m}-\sqrt{25m^6·2m}\\ & =& \sqrt{16m^6}·\sqrt{2m}-\sqrt{25m^6}·\sqrt{2m}\\ & =& 4m^3\sqrt{2m}-5m^3\sqrt{2m}\end{array}$

On combing the like radicals we get

$\begin{array}{rcl}4m^3\sqrt{2m}-5m^3\sqrt{2m}& =& (4-5)m^3\sqrt{2m}\\ & =& -m^3\sqrt{2m}\end{array}$

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

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

localid="1645681992780" $\begin{array}{rcl}\sqrt[3]{135x^7}-\sqrt[3]{40x^7}& =& \sqrt[3]{27x^6·5x}-\sqrt[3]{8x^6·5x}\\ & =& \sqrt[3]{27x^6}·\sqrt[3]{5x}-\sqrt[3]{8x^6}·\sqrt[3]{5x}\\ & =& 3x^2·\sqrt[3]{5x}-2x^2·\sqrt[3]{5x}\end{array}$

On combing the like radicals we get

localid="1645682007270" $$\begin{gathered} 3x^2·\sqrt[3]{5x}-2x^2·\sqrt[3]{5x}=(3-2)x^2\sqrt[3]{5x} \\ =x^2\sqrt[3]{5x} \end{gathered}$$