Simplification of Radical Expressions
Key Questions

Generally, you don't want to have radical at the denominators. So, let's say that we want to simplify the expression
#\frac{\sqrt{a}}{\sqrt{b}}# , where#a# and#b# can be any expression you want. Since, of course,#\frac{\sqrt{b}}{\sqrt{b}}=1# , we can multiply it without changing the value of our expression, so we have#\frac{\sqrt{a}}{\sqrt{b}}=\frac{\sqrt{a}}{\sqrt{b}} \cdot \frac{\sqrt{b}}{\sqrt{b}}# . The advantage is that now we observe that#\sqrt{b} \cdot \sqrt{b}=b# , and so our expression becomes#\frac\{\sqrt{ab}}{b}# , and we got rid of the radical at the denominator. 
Expression with a a square root , cubed root or other fractional exponents in the expression

This is easy! If you want to multiply this are the rules: First coefficients are multiplied with each other and the subradical amounts each other, placing the latter product under the radical sign common and the result is simplified.
Let's go:
#2sqrt5# times# 3sqrt10# #2sqrt5 × 3sqrt10 = 2 × 3sqrt(5×10)=6sqrt50# #= 6sqrt(2·5^2)# # = 30sqrt2# Now if you want to divide, then the coefficients are divided among themselves and subradical amounts each other, placing the latter quotient under the radical common and the result is simplified.
#2root3 (81x^7)# by#3root3( 3x^2)# #(2root3 (81x^7)) /(3root3 (3x^2)) = 2/3root3 ((81x^7)/(3x^2)) =2/3root3 (27x^5)# #2/3 root3 (3^3·x^3·x^2) = 2xroot3 (x^2)# I hope you can find it useful, and here is a link to solve this ones with different indices.

There are two common ways to simplify radical expressions, depending on the denominator.
Using the identities
#\sqrt{a}^2=a# and#(ab)(a+b)=a^2b^2# , in fact, you can get rid of the roots at the denominator.Case 1: the denominator consists of a single root. For example, let's say that our fraction is
#{3x}/{\sqrt{x+3}}# . If we multply this fraction by#{\sqrt{x+3}}/{\sqrt{x+3}}# , we won't change its value (since of course#{\sqrt{x+3}}/{\sqrt{x+3}}=1# , but we can rewrite it as follows:
#{3x}/{\sqrt{x+3}} \cdot {\sqrt{x+3}}/{\sqrt{x+3}}= \frac{3x\sqrt{x+3}}{\sqrt{x+3}^2}# , and finally obtain
#\frac{3x\sqrt{x+3}}{x+3}# Case 2: the denominator consists of a sum/difference of roots. If we multiply by the difference/sum of the roots, we'll have the same result as above. For example, if you have
#\frac{\cos(x)}{\sqrt{x}+\sqrt{\sin(x)}# You'll multiply numerator and denominator by the difference
#\sqrt{x}\sqrt{\sin(x)# , and obtain#\frac{\cos(x)}{\sqrt{x}+\sqrt{\sin(x))} \cdot \frac{\sqrt{x}\sqrt{\sin(x)}}{\sqrt{x}\sqrt{\sin(x)}}# which is#\frac{\cos(x)(\sqrt{x}\sqrt{\sin(x)})}{\sqrt{x}^2\sqrt{\sin(x)}^2}# which finally equals
#\frac{\cos(x) (\sqrt{x}\sqrt{\sin(x)})}{x\sin(x)}# Of course, when working with radicals, you always need to pay attention and make sure that the argument of the root is positive, otherwise you will write things that have no meaning!