To simplify the given expression, we can use the Quotient Property of Radicals to rewrite the given algebraic expression as the quotient of two radicals.
sqrt(27/m^5)=sqrt(27)/sqrt(m^5)
Now, we need to rationalize the denominator. To do that, we will multiply the numerator and the denominator by a factor that will make the expression inside the square root of the denominator a perfect square. We will use 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 that we've found the factors that will make the radicand of the denominator perfect squares 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
Because our radical has an even root and the variables have even exponents, we will need to use absolute value symbols to simplify our expression.
Next, we can simplify the denominator even further. To do so, we will use the fact that when the exponent of a variable inside the radical is odd and the simplified expression for that variable has an odd exponent, we do not need to use absolute value symbols.
sqrt(27)/sqrt(m^5) ⇔ 3sqrt(3m)/|m^3| = 3sqrt(3m)/m^3
