# How do you simplify sqrt(8x^3) +sqrt(50x^5)- sqrt(18x^3)- sqrt( 32x^5)?

Jun 2, 2018

See a solution process below:

#### Explanation:

First, we can rewrite each of the radicals as:

$\sqrt{4 {x}^{2} \cdot 2 x} + \sqrt{25 {x}^{4} \cdot 2 x} - \sqrt{9 {x}^{2} \cdot 2 x} - \sqrt{16 {x}^{4} \cdot 2 x}$

Now, use this rule for radicals to rewrite the radicals and factor out a common term and group then combine like terms:

$\sqrt{\textcolor{red}{a} \cdot \textcolor{b l u e}{b}} = \sqrt{\textcolor{red}{a}} \cdot \sqrt{\textcolor{b l u e}{b}}$

$\left(\sqrt{4 {x}^{2}} \cdot \sqrt{2 x}\right) + \left(\sqrt{25 {x}^{4}} \cdot \sqrt{2 x}\right) - \left(\sqrt{9 {x}^{2}} \cdot \sqrt{2 x}\right) - \left(\sqrt{16 {x}^{4}} \cdot \sqrt{2 x}\right) \implies$

$2 x \sqrt{2 x} + 5 {x}^{2} \sqrt{2 x} - 3 x \sqrt{2 x} - 4 {x}^{2} \sqrt{2 x} \implies$

$\left(2 x + 5 {x}^{2} - 3 x - 4 {x}^{2}\right) \sqrt{2 x} \implies$

$\left(2 x - 3 x + 5 {x}^{2} - 4 {x}^{2}\right) \sqrt{2 x} \implies$

((2 - 3)x + (5 - 4(x^2)sqrt(2x) =>

$\left(- 1 x + 1 {x}^{2}\right) \sqrt{2 x} \implies$

$\left(- x + {x}^{2}\right) \sqrt{2 x} \implies$

$\left({x}^{2} - x\right) \sqrt{2 x}$

We can now factor out another common term:

$\left(\left(x \cdot x\right) - \left(1 \cdot x\right)\right) \sqrt{2 x} \implies$

$\left(x - 1\right) x \sqrt{2 x}$