Question

Simplify:

1. $\left(2\sqrt{6y^4}\right)\left(12\sqrt{30y}\right)$
2. $(-4\sqrt[4]{9a^3})\left(3\sqrt[4]{27a^2}\right)$
   

Short answer

Part a.$\begin{array}{rcl}& & 144y^2\sqrt{5y}\end{array}$

Part b.$\begin{array}{rcl}& & -36a\sqrt[4]{3a}\end{array}$

Step-by-step solution

Part (a) Step 1. Multiply the radicals and simplify 

Using the product property of roots that is $\sqrt[n]{a}·\sqrt[n]{b}=\sqrt[n]{ab}$, the expression can be simplified to

$\begin{array}{rcl}\left(2\sqrt{6y^4}\right)\left(12\sqrt{30y}\right)& =& 2·12·(\sqrt{6y^4}·\sqrt{30y})\\ & =& 24\sqrt{6y^4·30y}\\ & =& 24\sqrt{180y^5}\end{array}$

Now, the radical can be simplified as

$\begin{array}{rcl}24\sqrt{180y^5}& =& 24\sqrt{36y^4·5y}\\ & =& 24\sqrt{36y^4}·\sqrt{5y}\\ & =& 24·6y^2\sqrt{5y}\\ & =& 144y^2\sqrt{5y}\end{array}$

Part (b) Step 1. Multiply the radicals and simplify 

Using the product property of roots that is $\sqrt[n]{a}·\sqrt[n]{b}=\sqrt[n]{ab}$, the expression can be simplified to

$\begin{array}{rcl}(-4\sqrt[4]{9a^3})\left(3\sqrt[4]{27a^2}\right)& =& (-4)·3·(\sqrt[4]{9a^3}·\sqrt[4]{27a^2})\\ & =& -12\sqrt[4]{9a^3·27a^2}\\ & =& -12\sqrt[4]{243a^5}\end{array}$

Now, the radical can be simplified as

$\begin{array}{rcl}-12\sqrt[4]{243a^5}& =& -12\sqrt[4]{81a^4·3a}\\ & =& -12\sqrt[4]{81a^4}·\sqrt[4]{3a}\\ & =& -12·3a·\sqrt[4]{3a}\\ & =& -36a\sqrt[4]{3a}\end{array}$