How can the properties of rational exponents be applied to simplify expressions with radicals or rational exponents?

Radicals can be also be written as numbers with rational exponents, which allows you to apply the same properties of rational exponents to them.

First, I will state something from the Law of Indices: ##a^(m/n)=root(n)(a^m)=(root(n)(a))^m##

Okay, for example we want to simplify this: ##root(3)(4) * root(6)(4)##

Because of the Law of Indices, we can write these as numbers with rational exponents. (To make it easier for us, let's write 4 as ##2^2##) ##root(3)(2^2) * root(6)(2^2)## ##=2^(2/3)*2^(2/6)## ##=2^(2/3)*2^(1/3)## (We simplified ##2^(2/6)## to ##2^(1/3)##)

Remember that when multiplying numbers with the same base, you simply add their exponents. In this case, both numbers have the same base (2). ##2^(2/3)*2^(1/3)## ##=2^((2+1)/3)## ##=2^(3/3)## ##=2^(1)## ##=2##

I'm not sure if this is the answer you are looking for, but I hope it helps!