To rationalize the denominator, we will multiply the numerator and denominator by a factor that will make the denominator a perfect square inside the square 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 squares in the denominator. Our goal is to have two of each factor.
Now, let's consider one of the most difficult parts of correctly simplifying radicals. When the exponent of a variable inside the radical is even and the simplified expression for that variable has an odd exponent, we need to use absolute value symbols.
ll
sqrt(x^2)= |x| & sqrt(x^3)= sqrt(x^2)sqrt(x)
& =xsqrt(x)
This is because we do not want the result to be negative — the range of a square root function is all real numbers greater than or equal to 0. Now, we can finally remove the variables from the radicals.
Finally, let's pay close attention to the given expression.
sqrt(68 ac^3)/sqrt(27a^2)
Note that the radicand of the denominator is always positive. For the radicand of the numerator to be positive, both variables a and c must have the same sign — both positive or both negative. We will use this fact to try to eliminate the absolute value symbol.
