Rationalizing Denominators With Two Terms

• In a previous modules we rationalized a single term denominator. We now turn our attention to rationalizing two term denominators.

2

3 5

• Example 1

Rationalize the denominator:

2

3 5

What should be multiplied times the denominator to eliminate the radical?

3 3 5 3 3 3 5

3 5 3

The is still a radical in the result!

2

3 5

Try something a little different.

3 5 3 5

3 3 5 3 5 3 5 5

3 3 5 3 5 3 5 5

3 5 3 5 3 25

3 25

22

The radical has been eliminated!

The pattern we used was the original denominator

3 5 3 5

… times the same two term expression, except for the opposite sign.

These two expressions are called conjugates.

• To rationalize a two term denominator multiply both numerator and denominator by the conjugate of the denominator.

2

3 5

• Back to our original problem - rationalize the denominator

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

2

3 53 5

3 5

2 3 5

22

Based on earlier work, we get …

3 5

11

11

3 5

11

5 3

11

or

• Example 2

Rationalize the denominator:

2 3

5 2

5 2

5 2

2 3 5 2

5 2 5 2

• Numerators:

2 3 5 2

2 5 2 2 3 5 3 2

2 5 2 2 15 6

• Denominators:

5 2 5 2

5 5 5 2 2 5 2 2

5 10 10 2

3

• Putting it back together …

2 3 5 2

5 2 5 2

2 5 2 2 15 6

3