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Rationalizing the Denominator by Multiplying by a Conjugate

Rationalizing the denominator of a radical expression is a method used to eliminate radicals from a denominator. If the denominator is a binomial with a rational part and an irrational part, then you'll need to use the conjugate of the binomial.

Binomials of the form a b + c b and a b c b are called conjugates. For example, 4 + 3 and 4 3 are conjugates.

The product of two conjugates results in a difference of two squares.

( 4 + 3 ) ( 4 3 ) = 4 2 ( 3 ) 2 = 16 3 or 13

Example:

Simplify.

3 5 2

Multiply both the numerator and denominator by the conjugate of the denominator.

3 5 2 = 3 5 2 5 + 2 5 + 2

The denominator is now a difference of squares .

= 3 ( 5 + 2 ) 5 2 ( 2 ) 2

Use the power of a product property in the denominator.

= 15 + 3 2 25 2

= 15 + 3 2 23

 

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