Multiply the values under each radical sign. For instance, √x * √y = √xy
Add any exponents if you are multiplying bases that are alike. For example, if you have √x^3 * √x^7, you would add the 3 and the 7 to get √x^10. You could also state this answer as (x^1/2)^10, which reduces to x^5 (since a square root is equal to that value to the 1/2 power). If you multiply bases that are not alike, you cannot add the exponents. For example, √x^3 * √y^7 simply equals √x^3*y^7 (with the radical sign extending over the entire phrase).
Multiply the numerator and denominator of a fraction by the denominator when you are dividing by a variable under a radical sign. For example, if you have √a /√b, multiply the top and bottom of the fraction by √b. This gives you √ab/b (since √b times √b equals b).
Divide like bases with exponents by subtracting the denominator's exponent from the numerator's exponent. For example, if you have √x^6 / √x^3, you can simplify this to √x^(6-3), which equals √x^3.