Problems on Irrational Numbers

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 (‘a’ and ‘b’) are to be compared, such that ‘a’ is greater than ‘b’, then a$$^{2}$$ will be greater than b$$^{2}$$ and a$$^{3}$$ will be greater than b$$^{2}$$ and so on, i.e., n$$^{th}$$ power of ‘a’ will be greater than n$$^{th}$$ power of ‘b’.

The same concept is to be applied for the comparison between rational and irrational numbers.

So, now let’s have look at some problems given below:

1. Compare √11 and √21.

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,

(√11)$$^{2}$$ = √11 × √11 = 11.

(√21)$$^{2}$$ = √21 × √21 = 21.

Now it is easier to compare 11 and 21.

Since, 21 > 11. So, √21 > √11.

2. Compare √39 and √19.

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,

(√39)$$^{2}$$ = √39 × √39 = 39.

(√19)$$^{2}$$ = √19 × √19 = 19

Now it is easier to compare 39 and 19. Since, 39 > 19.

So,√39 > √19.

3. Compare $$\sqrt[3]{15}$$ and $$\sqrt[3]{11}$$.

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,

$$(\sqrt[3]{15})^{3}$$ = $$\sqrt[3]{15}$$ × $$\sqrt[3]{15}$$ × $$\sqrt[3]{15}$$ = 15.

$$(\sqrt[3]{11})^{3}$$ = $$\sqrt[3]{11}$$ × $$\sqrt[3]{11}$$ × $$\sqrt[3]{11}$$ = 11.

Since, 15 > 11. So, $$\sqrt[3]{15}$$ > $$\sqrt[3]{11}$$.

4. Compare 5 and √17.

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 × 5 = 25.

(√17)$$^{2}$$ = √17 x × √17 = 17.

Since, 25 > 17. So, 5 > √17.

5. Compare 4 and $$\sqrt[3]{32}$$.

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,

4$$^{3}$$= 4 × 4 × 4 = 64.

$$(\sqrt[3]{32})^{3}$$ = $$\sqrt[3]{32}$$ × $$\sqrt[3]{32}$$ × $$\sqrt[3]{32}$$ = 32.

Since, 64 > 32. So, 4 > $$\sqrt[3]{32}$$.

6. Rationalize $$\frac{1}{4 + \sqrt{2}}$$.

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,

$$\frac{1}{4 + \sqrt{2}} \times (\frac{4 - \sqrt{2}}{4 - \sqrt{2}})$$

⟹ $$\frac{4 - \sqrt{2}}{4^{2} - \sqrt{2^{2}}}$$

⟹ $$\frac{4 - \sqrt{2}}{16 - 2}$$

⟹ $$\frac{4 - \sqrt{2}}{14}$$

So the rationalized fraction is: $$\frac{4 - \sqrt{2}}{14}$$.

7. Rationalize $$\frac{2}{14 - \sqrt{26}}$$.

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,

$$\frac{2}{14 - \sqrt{26}} \times \frac{14 + \sqrt{26}}{14 + \sqrt{26}}$$

⟹ $$\frac{2(14 - \sqrt{26})}{14^{2} - \sqrt{26^{2}}}$$

⟹ $$\frac{2(14 - \sqrt{26})}{196 - 26}$$

⟹ $$\frac{2(14 - \sqrt{26})}{170}$$

So, the rationalized fraction is: $$\frac{2(14 - \sqrt{26})}{170}$$.

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

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. Fraction as a Part of Collection | Pictures of Fraction | Fractional

Feb 24, 24 04:33 PM

How to find fraction as a part of collection? Let there be 14 rectangles forming a box or rectangle. Thus, it can be said that there is a collection of 14 rectangles, 2 rectangles in each row. If it i…

2. Fraction of a Whole Numbers | Fractional Number |Examples with Picture

Feb 24, 24 04:11 PM

Fraction of a whole numbers are explained here with 4 following examples. There are three shapes: (a) circle-shape (b) rectangle-shape and (c) square-shape. Each one is divided into 4 equal parts. One…

3. Identification of the Parts of a Fraction | Fractional Numbers | Parts

Feb 24, 24 04:10 PM

We will discuss here about the identification of the parts of a fraction. We know fraction means part of something. Fraction tells us, into how many parts a whole has been

4. Numerator and Denominator of a Fraction | Numerator of the Fraction

Feb 24, 24 04:09 PM

What are the numerator and denominator of a fraction? We have already learnt that a fraction is written with two numbers arranged one over the other and separated by a line.