# What is sqrt(2x^3)*sqrt(6x^2)*sqrt(10x) in simplified form?

Jan 28, 2017

$\sqrt{2 {x}^{3}} \cdot \sqrt{6 {x}^{2}} \cdot \sqrt{10 x} = 2 {x}^{3} \sqrt{30}$

#### Explanation:

Note that if $a , b \ge 0$ then

$\sqrt{a} \sqrt{b} = \sqrt{a b}$

By extension, we find if $a , b , c \ge 0$ then:

$\sqrt{a} \sqrt{b} \sqrt{c} = \sqrt{a b} \sqrt{c} = \sqrt{a b c}$

Note also that if $a \ge 0$ then:

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

In our example, we will assume $x \ge 0$ in order that all of the original square roots are well defined Real numbers.

So in our example:

$\sqrt{2 {x}^{3}} \cdot \sqrt{6 {x}^{2}} \cdot \sqrt{10 x} = \sqrt{2 {x}^{3} \cdot 6 {x}^{2} \cdot 10 x}$

$\textcolor{w h i t e}{\sqrt{2 {x}^{3}} \cdot \sqrt{6 {x}^{2}} \cdot \sqrt{10 x}} = \sqrt{2 {x}^{3} \cdot 2 {x}^{3} \cdot 30}$

$\textcolor{w h i t e}{\sqrt{2 {x}^{3}} \cdot \sqrt{6 {x}^{2}} \cdot \sqrt{10 x}} = \sqrt{{\left(2 {x}^{3}\right)}^{2}} \sqrt{30}$

$\textcolor{w h i t e}{\sqrt{2 {x}^{3}} \cdot \sqrt{6 {x}^{2}} \cdot \sqrt{10 x}} = 2 {x}^{3} \sqrt{30}$

Footnote

It seems to be a common error to assume that:

$\sqrt{{x}^{2}} = x$

This does hold, but only if $x \ge 0$.

If we want to cover the case $x < 0$ too then we could write:

$\sqrt{{x}^{2}} = \left\mid x \right\mid$

In the given example, we can deduce that the case $x \ge 0$ is intended, since otherwise $\sqrt{2 {x}^{3}}$ and $\sqrt{10 x}$ could take imaginary values.