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In the previous topics of rational numbers we have learnt to solve the problems regarding the fractional numbers, i.e., the numbers that have real numbers in their denominators. But we havenβt seen much problems regarding those fractions which have irrational numbers in their denominator. Yet I the topic of rationalization we have seen few examples on how to rationalize denominators. Under this topic weβll see more problems regarding the calculations of rationalization of denominators. Below are given some examples on how to rationalize the complex denominators and proceed further to solve the problems involving these types of complex denominators:-
1. Rationalize 1β11.
Solution:
Since the given fraction has an irrational denominator, so we need to rationalize this and make it more simple. So, to rationalize this, we will multiply the numerator and denominator of the given fraction by root 11, i.e., β11.So,
1β11 Γ β11β11
βΉ β1111
So, the required rationalized form of the given denominator is:
β1111.
2. Rationalize 1β21.
Solution:
The given fraction has an irrational denominator. So, we need to make it simple by rationalizing the given denominator. To do so, weβll have to multiply and divide the given fraction by root 21, i.e., β21.So,
1β21Γ β21β21
βΉβ2121
So the required rationalized fraction is:
β2121
3. Rationalize 1β39.
Solution:
Since the given fraction has an irrational denominator in it. So, to make the calculations more easy we need to make it simple and hence we need to rationalize the denominator. To do so, weβll have to multiply both the numerator and denominator of the fraction with root 39, i.e., β39. So,
1β39Γ β39β39
βΉβ3939
So, the required rationalized fraction is:
β3939.
4. Rationalize 14+β10.
Solution:
The given fraction consists of irrational denominator. To make the calculations more simplified we will have to rationalize the denominator of the given fraction. To do so, weβll have to multiply both numerator and denominator by conjugate of the given denominator, i.e., 4ββ104ββ10. So,
14+β10Γ 4ββ104ββ10
βΉ4ββ1042ββ102
{(a+ b)(a-b) = (a)2 - (b)2}
βΉ4ββ1016β10
βΉ 4ββ106
So the required rationalized fraction is:
4ββ106.
5. Rationalize 1β6ββ5.
Solution:
Since the given fraction has irrational denominator in it. So, to make it more simplified we will have to rationalize the denominator of the given fraction. To do so, weβll have to multiply both numerator and denominator of the fraction by β6+β5β6+β5. So,
1β6ββ5Γ β6+β5β6+β5
βΉ β6+β5β62ββ52
{(a+ b)(a-b) = (a)2 - (b)2}
βΉ β6+β51
βΉ β6+β5
So, the required rationalized fraction is:
β6+β5
6. Rationalize 2β11ββ6.
Solution:
Since, the given fraction has irrational denominator in it which makes the calculations more complex. So, to make them more simplified weβll have to rationalize the denominator of the given fraction. To do so, weβll have to multiply both numerator and denominator of the given fraction with β11+β6β11+β6.
So,
2β11ββ6Γβ11+β6β11+β6
[(a + b)(a - b) = (a)2 - (b)2]
βΉ2Γ(β11+β6)β112ββ62
βΉ 2Γ(β11+β6)11β6
βΉ 2Γ(β11+β6)5
So, the required rationalized fraction is:
2Γ(β11+β6)5.
Irrational Numbers
Definition of Irrational Numbers
Representation of Irrational Numbers on The Number Line
Comparison between Two Irrational Numbers
Comparison between Rational and Irrational Numbers
Problems on Irrational Numbers
Problems on Rationalizing the Denominator
Worksheet on Irrational Numbers
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