Rationalizing a Denominator or Numerator

The binomials of the form and are called conjugates of each other. For example,

or

are conjugates of each other. Because conjugates are the sum and difference of the same two terms, their product is the difference of the squares of these terms (see Section: Special Product Formulas); that is,

• Remark that and are conjugates of each other.
• If the denominator of a fraction is of the form (or ), we can rationalize the denominator by multiplying the numerator and denominator of the fraction by the conjugate (or ). For example, Similarly - If the denominator of a fraction is we multiply the numerator and denominator of the fraction by and use the Sum of Cubes formula to get a denominator of (See <a href="#sec:Ch0-Special-product">Section: Special Product Formulas</a> for the special product formulas). - If the denominator of a fraction is we multiply the numerator and denominator of the fraction by and use the Difference of Cubes formula to get a denominator of (See <a href="#sec:Ch0-Special-product">Section: Special Product Formulas</a> for the special product formulas). For example: <div class="example"> Remove the square roots in the denominator <details> <summary>Solution</summary> We multiply top and bottom by giving </details> </div> - Sometimes we need to rationalize the numerator. This process is similar to rationalizing the denominator. <div class="example"> Rationalize the numerator of <details> <summary>Solution</summary> To rationalize the numerator, we multiply both the numerator and the denominator by the conjugate of the numerator, , which is :