Question

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

Answer 1

You can take one big root:
`sqrt(2*4x^(1+2)y^(3+7))=sqrt(2*4*x^3y^10)=(2*2^2x^3y^10)^(1/2)=`
`=2^(1/2)*2^(2*1/2)x^(3*1/2)y^(10*1/2)=`
where you used the fact that `sqrt(x)=x^(1/2)`
`=2xy^5sqrt(2x)`

Answer 2

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
`root(a)(b) xx root(a)(c) = root(a)(b xx c)`

So
`sqrt(2xy^3) * sqrt(4x^2y^7)`

`= sqrt( 8x^3y^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)`

`= color(red)(2xy^5) sqrt(2x)`