In this page rationalizing the denominator we are going to see some examples to show that how to rationalize the denominator in fraction.
Definition:
Some times the denominator of a ratio a/b may be a surd. In those ratios, the denominator can be made a rational number by common procedure. This procedure is known as rationalizing the denominator.
If the denominator is as follows |
Procedure |
(1) If the denominator is in the form of √a where a is a rational number. |
Then we have to multiply both the numerator and denominator by the same (√a). For example |
Explanation: In this question we have √5 in the denominator since it is rational we have to multiply the same with the numerator and denominator to rationalize the given fraction. So that we are getting √15/5 | |
(2) If the denominator is in the form of a + √b where a and b are rational numbers. |
Then we have to multiply both the numerator and denominator by its conjugate (a - √b). For example |
Explanation: In this question we have 3 + √5 in the denominator so we have to multiply by its conjugate that is 3 - √5 with both numerator and denominator. Now the denominator is like in the form (a+ b) (a-b). So we have simplified this using a² - b².By simplifying this we get (3-√5)/2 as answer. | |
(3) If the denominator is in the form of a - √b where a and b are rational numbers. |
Then we have to multiply both the numerator and denominator by its conjugate (a + √b). For example |
Explanation: In this question we have 3 - √2 in the denominator
so we have to multiply by its conjugate that is 3 + √2 with both
numerator and denominator. Now the denominator is like in the form (a+b) (a-b). So we have simplified this using a² - b².By simplifying this
we get (15+5√2)/7 as answer. | |
(4) If the denominator is in the form of √a + √b where a and b are rational numbers. |
Then we have to multiply both the numerator and denominator by its conjugate (√a - √b). For example |
Explanation: In this question we have √2 + √5 in the denominator
so we have to multiply by its conjugate that is √2 - √5 with both
numerator and denominator. Now the denominator is like in the form (a+b)
(a-b). So we have simplified this using a² - b².By simplifying this
we get (2√5-2√2)/3 as answer. |