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Solving Quadratic Equations by Completing the Square
Graphing Logarithmic Functions
Division Property of Exponents
Adding and Subtracting Rational Expressions With Like Denominators
Rationalizing the Denominator
Multiplying Special Polynomials
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Solving Linear Systems of Equations by Elimination
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Solving Linear Systems of Equations by Graphing
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Parallel and Perpendicular Lines
Simplifying Square Roots
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Adding and Subtracting Fractions
Solving Linear Equations
Inequalities in one Variable
Recognizing Polynomial Equations from their Graphs
Scientific Notation
Factoring a Sum or Difference of Two Cubes
Solving Nonlinear Equations by Substitution
Solving Systems of Linear Inequalities
Arithmetics with Decimals
Finding the Equation of an Inverse Function
Plotting Points in the Coordinate Plane
The Product of the Roots of a Quadratic
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Solving Quadratic Equations by Completing the Square
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Rationalizing the Denominator

A. What It Means to Rationalize the Denominator

In order that all of us doing math can compare answers, we agree upon a common conversation, or set of rules, concerning the form of the answers.

For instance, we could easily agree that we would not leave an answer in the form of 3 + 4, but would write 7 instead.

When the topic switches to that of radicals, those doing math have agreed that a RADICAL IN SIMPLE FORM will not (among other things) have a radical in the denominator of a fraction. We will all change the form so there is no radical in the denominator.

Now a radical in the denominator will not be something as simple as . Instead, it will have a radicand which will not come out from under the radical sign like .

Since is an irrational number, and we need to make it NOT irrational, the process of changing its form so it is no longer irrational is called RATIONALIZING THE DENOMINATOR.

B. There are 3 Cases of Rationalizing the Denominator

1. Case I : There is ONE TERM in the denominator and it is a SQUARE ROOT.

2. Case II : There is ONE TERM in the denominator, however, THE INDEX IS GREATER THAN TWO. It might be a cube root or a fourth root.

3. Case III : There are TWO TERMS in the denominator.

Let's study Case III:

3. Case III : There are TWO TERMS in the denominator.

Example :

Procedure : We will multiply both top and bottom by the conjugate. The conjugate is the same two terms but with a different sign between them.

Since the conjugate for this numerator is , we will multiply top and bottom by that number.

Note how our use of the conjugate on the bottom is recreating the special product of The Product of Conjugate Binomials, as in: (x + 3)(x - 3) Which will always result in squaring everything to get x 2- 9. And, squaring everything would get rid of the term with the square root.

You might need to review the product of two conjugate binomials if you can't see how we got the denominator.

Note: We could distribute the square root of three in the numerator, but there doesn't seem to be any advantage in doing so. Also, we don't have any factors we can cancel, so this is the answer.

 

Review of the Product of the Conjugate Binomials

(x + 3)(x - 3) is the product of the conjugate binomials because we see the same two terms with different signs between them.

This could also be called "The product of the sum of two terms and the difference of the two terms."

This, then, would be the sum of x and 3 times the difference of x and 3.

The product will always be:

The Square of the First Term Minus the Square of the Second Term.

In this example, x 2 - 9 will be our answer.

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