How do you simplify (2sqrt27)times(3 sqrt32)?

Mar 24, 2017

$72 \sqrt{6}$

Explanation:

$\left(\textcolor{b l u e}{2} \sqrt{\textcolor{g r e e n}{27}}\right) \times \left(\textcolor{red}{3} \sqrt{\textcolor{m a \ge n t a}{32}}\right)$

$\textcolor{w h i t e}{\text{XXX}} = \textcolor{b l u e}{2} \times \textcolor{red}{3} \times \sqrt{\textcolor{g r e e n}{{3}^{3}}} \times \sqrt{\textcolor{m a \ge n t a}{{2}^{5}}}$

$\textcolor{w h i t e}{\text{XXX}} = 6 \times \textcolor{g r e e n}{3} \sqrt{\textcolor{g r e e n}{3}} \times \textcolor{m a \ge n t a}{{2}^{2}} \sqrt{\textcolor{m a \ge n t a}{2}}$

$\textcolor{w h i t e}{\text{XXX}} = 6 \times \textcolor{g r e e n}{3} \times \textcolor{m a \ge n t a}{{2}^{2}} \times \sqrt{\textcolor{g r e e n}{3} \times \textcolor{m a \ge n t a}{2}}$

$\textcolor{w h i t e}{\text{XXX}} = 72 \sqrt{6}$

Mar 24, 2017

$72 \sqrt{6}$

Explanation:

Using the $\textcolor{b l u e}{\text{law of radicals}}$

$\textcolor{red}{\overline{\underline{| \textcolor{w h i t e}{\frac{2}{2}} \textcolor{b l a c k}{\sqrt{a b} \Leftrightarrow \sqrt{a} \times \sqrt{b}} \textcolor{w h i t e}{\frac{2}{2}} |}}}$

To simplify the radicals consider the product of their factors of which one should be a $\textcolor{b l u e}{\text{perfect square}}$

$\Rightarrow \sqrt{27} = \sqrt{\textcolor{red}{9} \times 3} = \sqrt{\textcolor{red}{9}} \times \sqrt{3} = 3 \sqrt{3}$

$\Rightarrow \sqrt{32} = \sqrt{\textcolor{red}{16} \times 2} = \sqrt{\textcolor{red}{16}} \times \sqrt{2} = 4 \sqrt{2}$

$\Rightarrow 2 \sqrt{27} \times 3 \sqrt{32}$

$= 2 \times \left(3 \times \sqrt{3}\right) \times 3 \times \left(4 \times \sqrt{2}\right)$

$= \left(2 \times 3 \times 3 \times 4\right) \times \left(\sqrt{3} \times \sqrt{2}\right)$

$= 72 \sqrt{3 \times 2} = 72 \sqrt{6}$

Mar 24, 2017

$72 \sqrt{6}$

Explanation:

$\left(2 \sqrt{27}\right) \times \left(3 \sqrt{32}\right)$

$\therefore = 2 \sqrt{3 \cdot 3 \cdot 3} \times 3 \sqrt{2 \cdot 2 \cdot 2 \cdot 2 \cdot 2}$

$\therefore = 2 \times 3 \sqrt{3} \times 3 \cdot 2 \cdot 2 \sqrt{2}$

$\therefore = 6 \sqrt{3} \times 12 \sqrt{2}$

$\therefore = 72 \sqrt{3} \sqrt{2}$

$\therefore = 72 \sqrt{2 \cdot 3}$

$\therefore = 72 \sqrt{6}$