Rationalizing the Denominator

1. Rationalizing the Denominator

It is considered bad practice to have a radical in the denominator of a fraction. When this happens we multiply the numerator and denominator by the same thing in order to clear the radical. In the lesson on dividing radicals, we talked about how this was done with monomials. Here we will look at how this is done with binomials.

If the binomial is in the numerator the process to rationalize the denominator is essentially the same as with monomials The only difference is we will have to distribute in the numerator.

EXAMPLE

	Want to clear in denominator; Multiply both numerator and denominator by to clear the radical in the denominator
	Distribute the through the numerator, multiply the denominator by
	Evaluate any multiplication, ()
	Simplify radicals in numerator ( can be broken down to ), multiply out denominator
	Take square root of 9
	Reduce each term by 3
	Our Solution

hint

It is important to remember that when reducing the fraction we cannot reduce with just the 3 and 12 or just the 9 and 12. When we have addition or subtraction in the numerator or denominator we must divide all terms by the same number. As we are rationalizing it will always be important to constantly check our problem to see if it can be simplified more. We ask ourselves, can the fraction be reduced? Can the radicals be simplified? These steps may happen several times on our way to the solution.

2. Using a Conjugate

If the binomial occurs in the denominator we will have to use a different strategy to clear the radical.

EXAMPLE

Consider . If we were to multiply the denominator by we would have to distribute it and we would end up with . We have not cleared the radical, only moved it to another part of the denominator. So our current method will not work.

Instead, we will use what is called a conjugate. A conjugate is made up of the same terms, with the opposite sign in the middle. So for our example with in the denominator, the conjugate would be

The advantage of a conjugate is when we multiply them together, we have , which is a difference and a sum. If we multiply these, we get a difference of squares. The final value ends up being the square of and the square of 5, with subtraction in the middle:

big idea

To rationalize a denominator containing a radical expression, multiply the fraction using its conjugate. The product will no longer contain a radical.

EXAMPLE

	Want to clear radical from denominator; Multiply numerator and denominator by conjugate,
	Distribute numerator, difference of squares in denominator
	Evaluate multiplication in numerator and denominator
	Evaluate denominator
	Simplify solution by dividing by 2
	Our Solution

We could have reduced by dividing by -2 instead of 2, giving . Both answers are correct.

EXAMPLE

	Want to clear radicals from denominator; Multiply numerator and denominator by conjugate,
	Distribute numerator, difference of squares in denominator
	Evaluate multiplication in both numerator and denominator
	Simplify denominator
	Break down radicals: can be broken down to , can be broken down to
	Take square roots where possible
	Our Solution

The same process can be used when there is a binomial in the numerator and denominator. We just need to remember to FOIL out the numerator.

EXAMPLE

	Want to clear radicals from denominator; Multiply numerator and denominator by conjugate,
	FOIL in numerator, difference of squares in denominator
	Evaluate multiplication in numerator and denominator
	Simplify denominator
	Numerator cannot be simplified any further; Divide each term by denominator, 1
	Our Solution

EXAMPLE

	Want to clear radicals from denominator; Multiply numerator and denominator by conjugate,
	FOIL in numerator, difference of squares in denominator
	Evaluate multiplication in numerator and denominator
	Evaluate multiplication in denominator
	Evaluate subtraction in denominator; Cannot be simplified any further
	Our solution

did you know

During the 5th century BC in India, Aryabhata published a treatise on astronomy. His work included a method for finding the square root of numbers that have many digits.

term to know

Conjugate

The conjugate of a binomial is a binomial with the opposite sign between its terms.

summary

Rationalizing the denominator involves multiplying by a conjugate in both the denominator and the numerator of a fraction and then simplifying. The reason that we do that is because having an irrational radical in the denominator of a fraction is not considered simplified. The conjugate of the is just We just use the opposite sign in between the two terms. The conjugate of just a radical by itself is that same radical.