For any real number a, we have that sqrt(a^n)=a if n is an odd number, and that sqrt(a^n)=|a| if n is even.
sqrt(28b^2x^3)/2|b|
Practice makes perfect
To simplify the given expression, we can first rewrite the radical as a quotient of two radicals.
sqrt(7x^3/4b^2)=sqrt(7x^3)/sqrt(4b^2)
Now, we can rationalize the denominator. To do it, 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 non-negative. Otherwise, the radicals would be imaginary. With this in mind, let's consider the possible values of the variables x and b.
In the numerator, the index is even and the exponent of x is odd. Therefore, in order for this radical expression to result in a real number, x must be non-negative.
In the denominator, the index is even and the exponent of b is even. Therefore, the expression will be real whether the value of b is positive or negative.
This means that if we remove x from the radical, we will not need absolute value symbols. However, we would need them if we removed b from the radical.
