In order to simplify square roots with variables that are to be multiplied, treat the variables as factors. Once the radicands are factored completely, cancel any like terms and multiply the simplified square roots.
- Factor the radicands
Factor the radicands of the square roots to prime numbers, and write out the powers of the variables. For the equation 2 * sqrt(4 * x^5 * y) * sqrt(9 * x^3 * y), the factored out version is 2 * sqrt(2 * 2 * x * x * x * x * x * y) * sqrt(3 * 3 * x * x * x * y).
- Group any like terms together
In the factored square roots, group any two like terms together and bring them outside of the square root as a single term. The equation 2 * sqrt(2 * 2 * x * x * x * x * x * y) * sqrt(3 * 3 * x * x * x * y) becomes 2 * 2 * x * x * sqrt(xy) * 3 * x * sqrt(xy).
- Multiply out the factored equation
Once the equation is completely factored and the like terms are grouped, multiply it out: 2 * 2 * x * x * sqrt(xy) * 3 * x * sqrt(xy) = 4x^2 * sqrt(xy) * 3x * sqrt(xy) = 12x^3 * sqrt(x^2 * y^2) = 12x^5 * y^2.