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

9th Grade Math

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