# How do you multiply xsqrt(10x)*7sqrt(15x)?

Mar 3, 2018

The result is $35 {x}^{2} \sqrt{6}$.

#### Explanation:

Multiplication: $\sqrt{a} \cdot \sqrt{b} = \sqrt{a b}$

Simplification: $\sqrt{{a}^{2}} = a$

For this problem, first, multiply the radicals (in blue) and their coefficients (in red) together:

$\textcolor{w h i t e}{=} \textcolor{red}{x} \textcolor{b l u e}{\sqrt{10 x}} \cdot \textcolor{red}{7} \textcolor{b l u e}{\sqrt{15 x}}$

$= \textcolor{red}{x} \cdot \textcolor{b l u e}{\sqrt{10 x}} \cdot \textcolor{red}{7} \cdot \textcolor{b l u e}{\sqrt{15 x}}$

$= \textcolor{red}{x} \cdot \textcolor{red}{7} \cdot \textcolor{b l u e}{\sqrt{10 x}} \cdot \textcolor{b l u e}{\sqrt{15 x}}$

$= \textcolor{red}{7 x} \cdot \textcolor{b l u e}{\sqrt{10 x}} \cdot \textcolor{b l u e}{\sqrt{15 x}}$

$= \textcolor{red}{7 x} \cdot \textcolor{b l u e}{\sqrt{10 x \cdot 15 x}}$

$= \textcolor{red}{7 x} \cdot \textcolor{b l u e}{\sqrt{10 \cdot 15 \cdot x \cdot x}}$

$= \textcolor{red}{7 x} \setminus \cdot \textcolor{b l u e}{\sqrt{150 \cdot x \cdot x}}$

$= \textcolor{red}{7 x} \cdot \textcolor{b l u e}{\sqrt{150 \cdot {x}^{2}}}$

Next, use the multiplication rule backwards:

$\textcolor{w h i t e}{=} \textcolor{red}{7 x} \cdot \textcolor{b l u e}{\sqrt{150 \cdot {x}^{2}}}$

$= \textcolor{red}{7 x} \cdot \textcolor{b l u e}{\sqrt{150}} \cdot \textcolor{b l u e}{\sqrt{{x}^{2}}}$

Now, use the simplification rule:

$\textcolor{w h i t e}{=} \textcolor{red}{7 x} \cdot \textcolor{b l u e}{\sqrt{150}} \cdot \textcolor{b l u e}{\sqrt{{x}^{2}}}$

$= \textcolor{red}{7 x} \cdot \textcolor{b l u e}{\sqrt{150}} \cdot \textcolor{red}{x}$

$= \textcolor{red}{7 x} \cdot \textcolor{red}{x} \cdot \textcolor{b l u e}{\sqrt{150}}$

$= \textcolor{red}{7 {x}^{2}} \cdot \textcolor{b l u e}{\sqrt{150}}$

Technically, this answer is correct, but it can be simplified further by factoring $150$ and then using the simplification rule backward again:

$\textcolor{w h i t e}{=} \textcolor{red}{7 {x}^{2}} \cdot \textcolor{b l u e}{\sqrt{150}}$

$= \textcolor{red}{7 {x}^{2}} \cdot \textcolor{b l u e}{\sqrt{6 \cdot 25}}$

$= \textcolor{red}{7 {x}^{2}} \cdot \textcolor{b l u e}{\sqrt{6}} \cdot \textcolor{b l u e}{\sqrt{25}}$

$= \textcolor{red}{7 {x}^{2}} \cdot \textcolor{b l u e}{\sqrt{6}} \cdot \textcolor{b l u e}{\sqrt{{5}^{2}}}$

$= \textcolor{red}{7 {x}^{2}} \cdot \textcolor{b l u e}{\sqrt{6}} \cdot \textcolor{red}{5}$

$= \textcolor{red}{7 {x}^{2}} \cdot \textcolor{red}{5} \cdot \textcolor{b l u e}{\sqrt{6}}$

$= \textcolor{red}{35 {x}^{2}} \cdot \textcolor{b l u e}{\sqrt{6}}$