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:,