#### Ways to Rationalizing Binomial Denominators, explanation and examples about Rationalizing Binomial Denominators.

**Rationalizing Binomial Denominators;**

lf we expand we get:

and the expansion of gives us:

Notice that in each of these examples the results are rational numbers, 1 and 6. In general, if we expand a pair of binomials, or expressions with two terms, of the form where a and b are rational numbers we obtain :

which is a rational number since the two irrational parts of the expansion, and , sum to zero.

We call a pair of binomials of the form conjugates.

So and are a pair of conjugates.

Also and are a pair of conjugates.

We say that is the conjugate of and that is the conjugate of .

Also and are conjugates of each other.

We can use the fact that, when we expand a pair of conjugates of the form we always obtain a rational number, to rationalize the denominators of fractions having irrational binomial denominators. By multiplying both the numerator and denominator of the fraction by the conjugate of the denominator we obtain a rational denominator.

**Example 1 :** Rationalize the denominator and simplify :

Multiplying the numerator and denominator by the conjugate of the denominator:

**Example 2 :** Rationalize the denominator and simplify :

Simplifying each irrational number first:

Multiplying the numerator and denominator by the conjugate of the denominator:,