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(10xy^3)/2y
Practice makes perfect
Usually before dividing the given radical expressions, we need to answer two questions.
Can the expressions be divided?
If so, do absolute value symbols need to be added to the answer?
However, here we can already see that the expressions can be divided, so let's focus on the second question. Firstly, let's notice that the radical can be split into two radicals, one in the numerator and one in the denominator.
sqrt(5x/8y)=sqrt(5x)/sqrt(8y)Now, to answer the second question, consider the rule regarding absolute value symbols for any real number a.
sqrt(a^n)=
a if n is odd
|a| if n is even
Since both radicals are real numbers and index of 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 x and y.
In the numerator, the index is even and the exponent of x is odd. For the expression to result in a real number, the value of x must be positive.
In the denominator, the index is even and the exponent of y is odd. For the expression to result in a real number, the value of y must also be positive.
Since both variables can only take positive values, we do not need absolute value symbols.
Because there are not any perfect 4^(th) powers under the radical, we just need to rationalize the denominator. To do that, we can multiply the numerator and denominator by a factor that will create a perfect 4^(th) power. Let's start by finding the necessary exponents. Our goal is to have four of each factor.
