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Rationalizing a Binomial Denominator with Radicals

There is an unspoken law in math that a radical cannot be left in the denominator. The process of eliminating the radical from the denominator is called rationalizing. When the denominator is a binomial (two terms) the conjugate of the denominator has to be used to rationalize.

Let's start be reviewing conjugate.

3+ 2 is a binomial with a radical

3- 2 the conjugate (change the sign in the middle)


Example 1

  • 4 5 -3


= 4 ( 5 -3 ) . ( 5 +3 ) ( 5 +3 ) multiply the numerator and denominator by the conjugate of the     denominator

= 4 5 +12 5+3 5 -3 5 -9  use the distributive property to simplify the top and bottom

= 4 5 +12 -4 combine like terms and notice that by multiplying by the conjugate      that radicals are eliminated in the denominator

= 4 5 -4 + 12 -4 prepare to reduce fractions

= - 5 -3 reduce fractions

Or

=  -3- 5 answer written in equivalent a+bi form

Example 2

  •   2+2 3- 2


=   ( 2+ 2 ) ( 3- 2 ) . ( 3+ 2 ) ( 3+ 2 ) multiply the numerator and denominator by the conjugate of the        denominator

=   6+2 2 +3 2 +2 9+3 2 -3 2 -2   use the distributive property to simplify the top and bottom

= 8+5 2 7   combine like terms and notice that by multiplying by the conjugate       that radicals are eliminated in the denominator

Or

= 8 7 + 5 2 7 answer written in equivalent a+bi form

To rationalize a radical expression, multiply the numerator and the denominator by the conjugate of the denominator. The conjugate of a binomial is obtained by changing the middle sign to its opposite.



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