How do you multiply sqrt(2xy^3)*sqrt(4x^2y^7)?

Mar 24, 2015

You can take one big root:
$\sqrt{2 \cdot 4 {x}^{1 + 2} {y}^{3 + 7}} = \sqrt{2 \cdot 4 \cdot {x}^{3} {y}^{10}} = {\left(2 \cdot {2}^{2} {x}^{3} {y}^{10}\right)}^{\frac{1}{2}} =$
$= {2}^{\frac{1}{2}} \cdot {2}^{2 \cdot \frac{1}{2}} {x}^{3 \cdot \frac{1}{2}} {y}^{10 \cdot \frac{1}{2}} =$
where you used the fact that $\sqrt{x} = {x}^{\frac{1}{2}}$
$= 2 x {y}^{5} \sqrt{2 x}$

Mar 24, 2015

Remember that if the exponents of two radicals are equal the arguments of the radicals can be multiplied under the same radical exponent.
That is
$\sqrt[a]{b} \times \sqrt[a]{c} = \sqrt[a]{b \times c}$

So
$\sqrt{2 x {y}^{3}} \cdot \sqrt{4 {x}^{2} {y}^{7}}$

$= \sqrt{8 {x}^{3} {y}^{10}}$

In this particular example some roots can be extracted:
sqrt(8x^3y^10) = sqrt(color(red)(2)^2(2)* (color(red)(x)^2) (x) * (color(red)(y^5)^2)

$= \textcolor{red}{2 x {y}^{5}} \sqrt{2 x}$