Rationalize a Binomial Denominator

Rationalizing a Binomial Denominator

When a denominator has two terms, the goal is to remove the radical or other irrational part from the denominator by multiplying by a suitable form of 1. The key idea is to use the conjugate of the binomial.

1) Find the conjugate

If the denominator is of the form a + b, its conjugate is a - b. If it is a - b, its conjugate is a + b.

2) Multiply numerator and denominator

Multiply both the top and bottom by the conjugate. This does not change the value because you are multiplying by 1.

3) Use the difference of squares

The denominator becomes (a + b)(a - b) = a^2 - b^2. This removes the binomial and usually eliminates any radical from the denominator.

4) Simplify completely

Expand the numerator if needed, then reduce any common factors.

Check your work

Your final answer should have a simplified denominator with no binomial or radical left in the denominator. Also, make sure every factor was multiplied correctly in both numerator and denominator.