Rationalizing the Denominator
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Review Rationalizing Denominator with Square Roots
A series of free, online or Algebra II lessons.
Examples, videos, solutions, worksheets, and activities to help Algebra students.
In this lesson, we will learn
- how to rationalize denominators with square roots
- how to rationalize denominators with higher roots
- how to rationalize denominators with a binomial
- how to solve radical equations
Rationalizing the Denominator
Rationalizing the denominator is an algebraic technique used to eliminate radical expressions (like square roots, cube roots, etc.) from the denominator of a fraction. The goal is to rewrite the fraction in an equivalent form where the denominator is a rational number (an integer or a fraction without radicals).
The following diagram how to rationalize the denominator using a conjugate when necessary. Scroll down the page for more examples and solutions rationalizing the denominator.
Radicals Worksheets
Practice your skills with the following worksheets:
Printable & Online Radicals Worksheets
How to Rationalize the Denominator
The method depends on the type of radical expression in the denominator.
Case 1: The denominator is a single square root
Method: Multiply both the numerator and the denominator by the radical in the denominator.
Explanation: Since \(\sqrt{a} \times \sqrt{a} = a\), this eliminates the radical. Remember that multiplying the top and bottom by the same non-zero value doesn’t change the value of the fraction.
Case 2: The denominator is a binomial with square roots (e.g. \(\sqrt{a}\pm \sqrt{b}\))
Method: Multiply both the numerator and the denominator by the conjugate of the denominator.
Explanation: The conjugate of a binomial (A + B) is (A − B), and vice versa. When you multiply a binomial by its conjugate, you use the “difference of squares” formula: (A + B)(A − B)=A2 − B2. This eliminates the radicals.
Videos
How to rationalize a denominator?
Rationalizing the Denominator with Higher Roots
When a denominator has a higher root, multiplying by the radicand will not remove the root. Instead, to rationalize the denominator we multiply by a number that will yield a new term that can come out of the root. For example, with a cube root multiply by a number that will give a cubic number such as 8, 27, or 64.
How to rationalize the denominator with a higher root.
Rationalize Denominators - Monomial Higher Roots
Rationalize the denominator containing a cube root
Rationalizing a Denominator with a Binomial
When rationalizing a denominator with two terms, called a binomial, first identify the conjugate of the binomial. The conjugate is the same binomial except the second term has an opposite sign. Next, multiply the numerator and denominator by the conjugate. The denominator becomes a difference of squares, which will eliminate the square roots in the denominator.
How to rationalize a denominator by multiplying by the conjugate.
There will be times when you will need to rationalize the denominator and the denominator consists of a binomial radical. You need to multiply by the denominator’s conjugate.
Rationalizing Binomial Denominators Part 1
Rationalizing Binomial Denominators Part 2
Solving an Equation with Radicals
Solving equations with radicals, no matter what power, involves isolating the radical on one side of the equation and then raising both sides of the equation to the power of the radical. When solving radicals, the final step is to isolate the variable. If there are more than one radical, we isolate and remove one root, then isolate and remove the other root. Finally, we solve the remaining equation for the variable.
How to solve an equation with a square root.
Solving Radical Equations 1
Solving Radical Equations 2
Solving Radical Equations 3
Solving Radical Equations 4
Solving Radical Equations 5
Example of solving a Radical Equation that becomes a quadratic equation.
Shows how to solve a radical equation that has 2 radicals that after squaring both sides twice to eliminate both radicals produces a quadratic equation.
Shows how to solve a radical equation that has 2 radicals that after squaring both sides twice to eliminate both radicals produces a quadratic equation.
Try out our new and fun Fraction Concoction Game.
Add and subtract fractions to make exciting fraction concoctions following a recipe. There are four levels of difficulty: Easy, medium, hard and insane. Practice the basics of fraction addition and subtraction or challenge yourself with the insane level.
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