# Worksheet on Irrational Numbers

From previous topics of irrational numbers it has become clear that rationalization of denominator is one of the most important steps done while doing calculations which involve irrational denominators. In the previous topic of rationalization we have learnt how to rationalize the denominator. In this topic, we will get to solve some problems regarding rationalization of denominators. Below are given some problems involving calculation of rationalization of denominator:

1. Rationalize $$\frac{1}{\sqrt{11}}$$.

2. Rationalize $$\frac{1}{\sqrt{37}}$$.

3. Rationalize $$\frac{1}{\sqrt{17}}$$.

4. Rationalize $$\frac{1}{\sqrt{23}}$$.

5. Rationalize $$\frac{1}{\sqrt{46}}$$.

6. Rationalize $$\frac{1}{\sqrt{37}}$$.

7. Rationalize $$\frac{1}{1+\sqrt{3}}$$.

8. Rationalize $$\frac{1}{1+\sqrt{7}}$$.

9. Rationalize $$\frac{1}{4+\sqrt{13}}$$.

10. Rationalize $$\frac{1}{7+\sqrt{29}}$$.

11. Rationalize $$\frac{1}{11-\sqrt{13}}$$.

12. Rationalize $$\frac{1}{9-\sqrt{57}}$$.

13. Rationalize $$\frac{1}{13-\sqrt{15}}$$.

14. Rationalize $$\frac{1}{\sqrt{13}-\sqrt{11}}$$.

15. Rationalize $$\frac{1}{\sqrt{21}-\sqrt{29}}$$.

16. Rationalize $$\frac{1}{\sqrt{31}+\sqrt{41}}$$.

17. Rationalize $$\frac{1}{\sqrt{21}+\sqrt{37}}$$.

18. Rationalize $$\frac{2}{\sqrt{5}+\sqrt{7}}$$.

19. Rationalize $$\frac{5}{\sqrt{28}+\sqrt{37}}$$.

20. Rationalize $$\frac{6}{\sqrt{53}-\sqrt{49}}$$.

21. Rationalize $$\frac{17}{\sqrt{53}-\sqrt{49}}$$.

22. Rationalize the denominator and find the conjugate of the fraction so formed- $$\frac{1}{\sqrt{5}-\sqrt{4}}$$.

23. Rationalize the denominator and find the conjugate of the resulting fraction- $$\frac{2}{\sqrt{11}-\sqrt{9}}$$.

24. Rationalize the fraction and find the conjugate of the resulting fraction- $$\frac{6}{\sqrt{21}-\sqrt{19}}$$.

25. Rationalize the given fraction and find the conjugate of the resulting fraction- $$\frac{10}{\sqrt{59}-\sqrt{41}}$$.

26. Rationalize the fraction and find the conjugate of the resulting fraction- $$\frac{19}{21-\sqrt{41}}$$.

27. Find the value of ‘a’ in the given equation:

$$\frac{1}{\sqrt{17}-\sqrt{15}}$$ = $$\frac{\sqrt{a}+\sqrt{15}}{2}$$

28. Find the value of ‘a’ in the given equation:

$$\frac{1}{\sqrt{19}-\sqrt{12}}$$ = $$\frac{\sqrt{19}+\sqrt{a}}{7}$$

29. Find the value of ‘a’ in the given equation:

$$\frac{2}{11+\sqrt{14}}$$ = \frac{2(11-\sqrt{14})}{a}\)

30. Solve the following problem:

$$\frac{1}{9+\sqrt{3}} + \frac{1}{3+\sqrt{2}}$$.

31. Solve the following arithematic:

$$\frac{2}{11+\sqrt{15}} + \frac{9}{2+\sqrt{8}}$$.

32. Solve the following:

$$\frac{11}{\sqrt{8}} + \frac{15}{\sqrt{21}}$$.

Solutions:

1. $$\frac{\sqrt{11}}{11}$$

2. $$\frac{\sqrt{37}}{37}$$

3. $$\frac{\sqrt{17}}{17}$$

4. $$\frac{\sqrt{23}}{23}$$

5. $$\frac{\sqrt{46}}{46}$$

6. $$\frac{\sqrt{71}}{71}$$

7. $$\frac{\sqrt{3}-1}{2}$$

8. $$\frac{\sqrt{7}-1}{6}$$

9. $$\frac{4-\sqrt{13}}{3}$$

10. $$\frac{7-\sqrt{29}}{20}$$

11. $$\frac{11+\sqrt{13}}{108}$$

12. $$\frac{9+\sqrt{57}}{24}$$

13. $$\frac{-13-\sqrt{15}}{2}$$

14. $$\frac{\sqrt{13}+\sqrt{11}}{2}$$

15. $$\frac{\sqrt{29}-\sqrt{21}}{8}$$

16. $$\frac{\sqrt{41}-\sqrt{31}}{10}$$

17. $$\frac{\sqrt{37}-\sqrt{21}}{16}$$

18. $$\frac{\sqrt{37}-\sqrt{21}}{16}$$

19. $$\frac{5(\sqrt{37}-\sqrt{28})}{9}$$

20. $$\frac{3(\sqrt{53}+7)}{2}$$

21. $$\frac{17(\sqrt{53}+7)}{4}$$

22. $$\frac{\sqrt{5}-\sqrt{4}}{1}$$

23. $$\frac{\sqrt{11}+\sqrt{9}}{1}$$

24. $$\frac{3(\sqrt{19}-\sqrt{21})}{1}$$

25. $$\frac{5(\sqrt{41}-\sqrt{59})}{9}$$

26. $$\frac{19(\sqrt{41}-21)}{400}$$

27. a = √17

28. a = √12

29. a = 107

30. $$\frac{-171-7\sqrt{3}-78\sqrt{2}}{546}$$

31. $$\frac{477\sqrt{2}-2\sqrt{15}-455}{106}$$

32. $$\frac{231+120\sqrt{21}}{168}$$

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

Rationalization

Problems on Irrational Numbers

Problems on Rationalizing the Denominator

Worksheet on Irrational Numbers

Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.

## Recent Articles

1. ### Types of Fractions |Proper Fraction |Improper Fraction |Mixed Fraction

Jul 12, 24 03:08 PM

The three types of fractions are : Proper fraction, Improper fraction, Mixed fraction, Proper fraction: Fractions whose numerators are less than the denominators are called proper fractions. (Numerato…

2. ### Worksheet on Fractions | Questions on Fractions | Representation | Ans

Jul 12, 24 02:11 PM

In worksheet on fractions, all grade students can practice the questions on fractions on a whole number and also on representation of a fraction. This exercise sheet on fractions can be practiced

3. ### Fraction in Lowest Terms |Reducing Fractions|Fraction in Simplest Form

Jul 12, 24 03:21 AM

There are two methods to reduce a given fraction to its simplest form, viz., H.C.F. Method and Prime Factorization Method. If numerator and denominator of a fraction have no common factor other than 1…

4. ### Conversion of Improper Fractions into Mixed Fractions |Solved Examples

Jul 12, 24 12:59 AM

To convert an improper fraction into a mixed number, divide the numerator of the given improper fraction by its denominator. The quotient will represent the whole number and the remainder so obtained…

5. ### Conversion of Mixed Fractions into Improper Fractions |Solved Examples

Jul 12, 24 12:30 AM

To convert a mixed number into an improper fraction, we multiply the whole number by the denominator of the proper fraction and then to the product add the numerator of the fraction to get the numerat…