To simplify the given expression, we can first rewrite the radical as a quotient of two radicals.
sqrt(12x^3y^2/5a^2b)=sqrt(12x^3y^2)/sqrt(5a^2b)
Now, we can rationalize the denominator of the quotient. To do that, we will multiply the numerator and denominator by a factor that will make the denominator a perfect 4^(th) power inside the 4^(th) root. We will do this using the fact that we can multiply the radicands of radicals if they have the same index.
If sqrt(a) and sqrt(b) are real numbers,
then sqrt(a)* sqrt(b)= sqrt(ab).
Let's start by finding the exponents necessary to create perfect 4^(th) powers in the denominator. Our goal is to have four of each factor.
Now that we have found the factors that will make the radicand of the denominator perfect 4^(th) powers only, we can begin to simplify the quotient. While simplifying, we should consider the index of the radicals to see how we should format our solution.
sqrt(a^n)=
a if n is odd
|a| if n is even
Since both radicals are real numbers and the roots are even, the expressions underneath the radicals are positive. Otherwise, the radicals would be imaginary. With this in mind, let's consider the possible values of the variables a, b, x, and y.
In the numerator, the index is even and the exponents of x and b are odd. Therefore, in order for this radical expression to result in a real number, x and b must be positive.
In the numerator, the index is even and the exponents of y and a are even. Therefore, the expression will be real whether the values of y and a are positive or negative.
This means that if we remove x or b from the radical, we will not need absolute value symbols. However, we would need them if we removed y or a from the radical.
