Subscribe to our βΆοΈ YouTube channel π΄ for the latest videos, updates, and tips.
Home | About Us | Contact Us | Privacy | Math Blog
Till here we have learnt many concepts regarding irrational numbers. Under this topic we will be solving some problems related to irrational numbers. It will contain problems from all topics of irrational numbers.
Before moving to problems, one should look at the basic concepts regarding the comparison of irrational numbers.
For comparing them, we should always keep in mind that if square or cube roots of two numbers (βaβ and βbβ) are to be compared, such that βaβ is greater than βbβ, then a2 will be greater than b2 and a3 will be greater than b2 and so on, i.e., nth power of βaβ will be greater than nth power of βbβ.
The same concept is to be applied for the comparison between rational and irrational numbers.
So, now letβs have look at some problems given below:
1. Compare β11 and β21.
Solution:
Since the given numbers are not the perfect square roots so the numbers are irrational numbers. To compare them let us first compare them into rational numbers. So,
(β11)2 = β11 Γ β11 = 11.
(β21)2 = β21 Γ β21 = 21.
Now it is easier to compare 11 and 21.
Since, 21 > 11. So, β21 > β11.
2. Compare β39 and β19.
Solution:
Since the given numbers are not the perfect square roots of any number, so they are irrational numbers. To compare them, we will first compare them into rational numbers and then perform the comparison. So,
(β39)2 = β39 Γ β39 = 39.
(β19)2 = β19 Γ β19 = 19
Now it is easier to compare 39 and 19. Since, 39 > 19.
So,β39 > β19.
3. Compare 3β15 and 3β11.
Solution:
Since the given numbers are not the perfect cube roots. So, to make comparison between them e first need to convert them into rational numbers and then perform the comparison. So,
(3β15)3 = 3β15 Γ 3β15 Γ 3β15 = 15.
(3β11)3 = 3β11 Γ 3β11 Γ 3β11 = 11.
Since, 15 > 11. So, 3β15 > 3β11.
4. Compare 5 and β17.
Solution:
Among the numbers given, one of them is rational while other one is irrational. So, to make comparison between them, we will raise both of them to them to the same power such that the irrational one becomes rational. So,
(5)2 = 5 Γ 5 = 25.
(β17)2 = β17 x Γ β17 = 17.
Since, 25 > 17. So, 5 > β17.
5. Compare 4 and 3β32.
Solution:
Among the given numbers to make comparison, one of them is rational while other one is irrational. So, to make comparison both numbers will be raised to the same power such that the irrational one becomes rational. So,
43= 4 Γ 4 Γ 4 = 64.
(3β32)3 = 3β32 Γ 3β32 Γ 3β32 = 32.
Since, 64 > 32. So, 4 > 3β32.
6. Rationalize 14+β2.
Solution:
Since the given fraction contains irrational denominator, so we need to convert it into a rational denominator so that calculations may become easier and simplified ones. To do so we will multiply both numerator and denominator by the conjugate of the denominator. So,
14+β2Γ(4ββ24ββ2)
βΉ 4ββ242ββ22
βΉ 4ββ216β2
βΉ 4ββ214
So the rationalized fraction is: 4ββ214.
7. Rationalize 214ββ26.
Solution:
Since the given fraction contains irrational denominator, so we need to convert it into a rational denominator so that calculations may become easier and simplified ones. To do so we will multiply both numerator and denominator by the conjugate of the denominator. So,
214ββ26Γ14+β2614+β26
βΉ 2(14ββ26)142ββ262
βΉ 2(14ββ26)196β26
βΉ 2(14ββ26)170
So, the rationalized fraction is: 2(14ββ26)170.
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
From Problems on Irrational Numbers to HOME PAGE
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.
Jul 28, 25 03:00 AM
Jul 25, 25 03:15 AM
Jul 24, 25 03:46 PM
Jul 23, 25 11:37 AM
Jul 20, 25 10:22 AM
New! Comments
Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.