NCERT Exemplar for Class 9 Math Chapter 1 - Number Systems (Book Solutions)

Sample Questions

1. Which of the following is not equal to $${\left[{\left({\displaystyle \frac{5}{6}}\right)}^{{\displaystyle \frac{1}{5}}}\right]}^{-{\displaystyle \frac{1}{6}}}$$ ?

(A) $${\left[\left({\displaystyle \frac{5}{6}}\right)\right]}^{\left({\displaystyle \frac{1}{5}}-{\displaystyle \frac{1}{6}}\right)}$$            

(B) ${\displaystyle \frac{1}{{\left[{\left({\displaystyle \frac{5}{6}}\right)}^{\left({\displaystyle \frac{1}{5}}\right)}\right]}^{\left({\displaystyle \frac{1}{6}}\right)}}}$          

(C) ${\left({\displaystyle \frac{6}{5}}\right)}^{\left({\displaystyle \frac{1}{30}}\right)}$             

(D) ${\left({\displaystyle \frac{5}{6}}\right)}^{\left({\displaystyle \frac{1}{30}}\right)}$ 

Ans: option b is correct

If $p$ and $$q$$ are rational numbers and $$a$$ is a positive real number, then

${\left(a^p\right)}^q={\left(a\right)}^{pq}$ . So, this law of exponents is to be used to find the value of a given expression.

$\Rightarrow {\left[{\left({\displaystyle \frac{5}{6}}\right)}^{{\displaystyle \frac{1}{5}}}\right]}^{-{\displaystyle \frac{1}{6}}}=\left[{\left({\displaystyle \frac{5}{6}}\right)}^{{\displaystyle \frac{1}{5}}\times \left({\displaystyle \frac{-1}{6}}\right)}\right]$

EXERCISE 1.1

1. Every rational number is 

(A) a natural number 

(B) an integer 

(C) a real number 

(D) a whole number

Ans: C

Real Number is the union of rational numbers and the irrational numbers. 

Hence, every rational number is a real number.

2. Between two rational numbers 

(A) There is no rational number 

(B) There is exactly one rational number

(C) There are infinitely many rational numbers

(D) There are only rational numbers and no irrational numbers

Ans: C

Rational number is just the division of any two integers. 

To find the rational number between two rational number, use the following steps:

• First, check the values of Denominators.

• If denominators are the same, then check the value of the numerator.

• If the numerators differ by a large value then write the rational numbers with an increment of one (keeping the denominator part unchanged.)

• If the numerators differ by a smaller value than the number of rational numbers to be found, simply multiply the numerators and denominators by multiples of 10.

Hence, we can find infinitely many rational numbers between two rational numbers.

3. Decimal representation of a rational number cannot be 

(A) terminating 

(B) non-terminating 

(C) non-terminating repeating

 (D) non-terminating non-repeating

Ans: D

A rational number is expressed in decimal form.

A rational number can be either terminating or non-terminating recurring decimal.

Hence, Decimal expansion of any rational number cannot be non-terminating non-repeating.

4. The product of any two irrational numbers is 

(A) Always an irrational number 

(B) Always a rational number 

(C) Always an integer 

(D) Sometimes rational, sometimes irrational

Ans: D

The product of any two irrational numbers can be rational or irrational.

Taking an example to illustrate the above statement,

Let $\left(4+\sqrt{5}\right),\left(4-\sqrt{5}\right)\text{ and }\left(3-\sqrt{5}\right)$ be rational numbers.

Consider, 

$\left(4+\sqrt{5}\right)\left(4-\sqrt{5}\right)$

 $=\left[{\left(4\right)}^2-{\left(\sqrt{5}\right)}^2\right]$

 $=16-5$

 $=11$                                                                   

 $\left(4+\sqrt{5}\right)\left(3-\sqrt{5}\right)$

 $=4\left(3-\sqrt{5}\right)+\sqrt{5}\left(3-\sqrt{5}\right)$

$=12-4\sqrt{5}+3\sqrt{5}-5$

$=7-\sqrt{5}$ 

The product is rational.                                               

Hence, the product of any two irrational numbers is sometimes rational, sometimes irrational

5. The decimal expansion of the number $\sqrt{2}$ is 

(A) A finite decimal

 (B) 1.41421 

(C) non-terminating recurring 

(D) non-terminating non-recurring

Ans: D

The given number $\sqrt{2}$ is an irrational number.

We know that:

When any rational number is expressed in the form of decimal expansion then it is either terminating or non-terminating recurring.

So, its expansion will be non- terminating, non- recurring.

6. Which of the following is irrational?

(A) $\sqrt{{\displaystyle \frac{4}{9}}}$                

(B) ${\displaystyle \frac{\sqrt{12}}{\sqrt{3}}}$                

(C) $\sqrt{7}$              

(D) $\sqrt{81}$ 

Ans: C

On solving all the options:

$\sqrt{{\displaystyle \frac{4}{9}}}={\displaystyle \frac{2}{3}}$ rational

${\displaystyle \frac{\sqrt{12}}{\sqrt{3}}}={\displaystyle \frac{2\sqrt{3}}{\sqrt{3}}}=2\text{ rational}$

$\sqrt{81}=9\text{ rational}$ 

$\sqrt{7}$ is an irrational number.

7. Which of the following is irrational?

(A) $$0.14$$            

(B) $$0.14\overline{16}$$             

(C) $0.\overline{1416}$            

(D) $$0.4014001400014...$$ 

Ans: D

The number $$0.4014001400014...$$ is non- terminating and non-recurring.

It can not be represented as a simple fraction.

Hence, the number $$0.4014001400014...$$ is irrational    

8. A rational number between $$\sqrt{2}$$  and $\sqrt{3}$  is

(A) $${\displaystyle \frac{\sqrt{2}+\sqrt{3}}{2}}$$                  

(B) $${\displaystyle \frac{\sqrt{2}-\sqrt{3}}{2}}$$                   

(C) $1.5$            

(D) $$1.8$$ 

Ans: C

We know that:

$\sqrt{2}=1.414$

$\sqrt{3}=1.732$ 

The number $1.5$ lies between 1.414 and 1.732.

Hence, a rational number between $$\sqrt{2}$$  and $\sqrt{3}$  is 1.5.

9. The value of $$1.999...$$ in the form ${\displaystyle \frac{p}{q}}$ where $$p$$ and $$q$$ are integers and $$q\ne 0,$$ is

(A) ${\displaystyle \frac{19}{10}}$                                    

(B) ${\displaystyle \frac{1999}{1000}}$                                      

(C) $2$                                   

(D) ${\displaystyle \frac{1}{9}}$ 

Ans: C

Assume $x=1.999......(1)$ 

Multiply equation (1) by 10, we get:

$10x=19.999........(2)$ 

Subtracting $$\left(1\right)$$ from $$\left(2\right)$$ 

$10x-x=\left(19.999...\right)-\left(1.999...\right)$

$9x=18$

$x={\displaystyle \frac{18}{9}}$

$x=2$ 

10. $2\sqrt{3}+\sqrt{3}$ is equal to

(A) $2\sqrt{6}$                                    

(B) $6$                                      

(C) $3\sqrt{3}$                                  

(D) $4\sqrt{6}$ 

Ans: C

Let $2\sqrt{3}+\sqrt{3}$

Taking $\sqrt{3}$ as common 

$2\sqrt{3}+\sqrt{3}$

$=\sqrt{3}\left(2+1\right)$

$=3\sqrt{3}$ 

Hence, $2\sqrt{3}+\sqrt{3}$= $3\sqrt{3}$

11.  $\sqrt{10}\times \sqrt{15}$ is equal to

(A) $6\sqrt{5}$                   

(B) $5\sqrt{6}$                  

(C) $\sqrt{25}$                  

(D) $10\sqrt{5}$

Ans: B

Let a and b be positive real numbers, then 

$\sqrt{a}\sqrt{b}=\sqrt{ab}$

$\sqrt{10}\times \sqrt{15}=\sqrt{10\times 15}$

$=\sqrt{2\times 5\times 3\times 5}$

$=\sqrt{25\times 6}$

$=5\sqrt{6}$

12. The number obtained on rationalizing the denominator of ${\displaystyle \frac{1}{\sqrt{7}-2}}$ is

(A) ${\displaystyle \frac{\sqrt{7}+2}{3}}$         

(B)  ${\displaystyle \frac{\sqrt{7}-2}{3}}$       

(C)  ${\displaystyle \frac{\sqrt{7}+2}{5}}$          

(D)  ${\displaystyle \frac{\sqrt{7}+2}{45}}$

Ans: A

To find the number obtained on rationalizing the denominator of ${\displaystyle \frac{1}{\sqrt{7}-2}}$ , multiply both numerator and denominator by $\sqrt{7}+2$

 ${\displaystyle \frac{1}{\sqrt{7}-2}}$$={\displaystyle \frac{1}{\sqrt{7}-2}}\times {\displaystyle \frac{\sqrt{7}+2}{\sqrt{7}+2}}$

$={\displaystyle \frac{\sqrt{7}+2}{\left(\sqrt{7}-2\right)\left(\sqrt{7}+2\right)}}$

$={\displaystyle \frac{\sqrt{7}+2}{{\left(\sqrt{7}\right)}^2-{\left(2\right)}^2}}....................(a+b)(a-b)=(a^2-b^2)$ 

 $={\displaystyle \frac{\sqrt{7}+2}{7-4}}$

 $={\displaystyle \frac{\sqrt{7}+2}{3}}$ 

Hence, the number obtained on rationalizing the denominator of ${\displaystyle \frac{1}{\sqrt{7}-2}}$ is ${\displaystyle \frac{\sqrt{7}+2}{3}}$. 

13.  ${\displaystyle \frac{1}{\sqrt{9}-\sqrt{8}}}$ is equal to

(A)  ${\displaystyle \frac{1}{2}}\left(3-2\sqrt{2}\right)$     

(B)   ${\displaystyle \frac{1}{3+2\sqrt{2}}}$       

(C)   $3-2\sqrt{2}$         

(D)   $3+2\sqrt{2}$

Ans: D  

To find the value of ${\displaystyle \frac{1}{\sqrt{9}-\sqrt{8}}}$, first we need to rationalize it. This can be done by multiplying both numerator and denominator by $\sqrt{9}+\sqrt{8}$                                  

$={\displaystyle \frac{1}{\sqrt{9}-\sqrt{8}}}$

$={\displaystyle \frac{1}{\sqrt{9}-\sqrt{8}}}\times {\displaystyle \frac{\sqrt{9}+\sqrt{8}}{\sqrt{9}+\sqrt{8}}}$

$={\displaystyle \frac{3+2\sqrt{2}}{9-8}}$

$={\displaystyle \frac{3+2\sqrt{2}}{1}}$

$=3+2\sqrt{2}$ 

The value of ${\displaystyle \frac{1}{\sqrt{9}-\sqrt{8}}}$  is $$3+2\sqrt{2}$$. 

14. After rationalizing the denominator of  $${\displaystyle \frac{7}{3\sqrt{3}-2\sqrt{2}}}$$, we get the denominator as 

(A) $$13$$                    

(B) $$19$$                  

(C) $$5$$                  

(D) $$35$$

Ans: B

To find the number obtained on rationalizing the denominator of $${\displaystyle \frac{7}{3\sqrt{3}-2\sqrt{2}}}$$ , multiply both numerator and denominator by $3\sqrt{3}+2\sqrt{2}$.

$${\displaystyle \frac{7}{3\sqrt{3}-2\sqrt{2}}}$$  

$={\displaystyle \frac{7}{3\sqrt{3}-2\sqrt{2}}}\times {\displaystyle \frac{3\sqrt{3}+2\sqrt{2}}{3\sqrt{3}+2\sqrt{2}}}$

$={\displaystyle \frac{7\left(3\sqrt{3}+2\sqrt{2}\right)}{{\left(3\sqrt{3}\right)}^2-{\left(2\sqrt{2}\right)}^2}}$   

$={\displaystyle \frac{7\left(3\sqrt{3}+2\sqrt{2}\right)}{27-8}}$

$={\displaystyle \frac{7\left(3\sqrt{3}+2\sqrt{2}\right)}{19}}$ 

Therefore, the denominator after rationalization is 19

15. The value of $${\displaystyle \frac{\sqrt{32}+\sqrt{48}}{\sqrt{8}+\sqrt{12}}}$$ is equal to

(A) $$\sqrt{2}$$               

(B) $$2$$              

(C) $$4$$                

(D)  $$8$$ 

Ans: B

$${\displaystyle \frac{\sqrt{32}+\sqrt{48}}{\sqrt{8}+\sqrt{12}}}$$                       

$={\displaystyle \frac{4\sqrt{2}+4\sqrt{3}}{2\sqrt{2}+2\sqrt{3}}}$

$={\displaystyle \frac{4\left(\sqrt{2}+\sqrt{3}\right)}{2\left(\sqrt{2}+\sqrt{3}\right)}}$

$={\displaystyle \frac{4}{2}}$

$=2$ 

Hence, the value of $${\displaystyle \frac{\sqrt{32}+\sqrt{48}}{\sqrt{8}+\sqrt{12}}}$$ is 2.

16. If $\sqrt{2}=1.4142$, then $\sqrt{{\displaystyle \frac{\sqrt{2}-1}{\sqrt{2}+1}}}$ is equal to

(A) $2.4142$         

(B) $5.8282$           

(C) $0.4142$          

(D) $0.1718$

Ans: C

$\sqrt{{\displaystyle \frac{\sqrt{2}-1}{\sqrt{2}+1}}}$

$=\sqrt{{\displaystyle \frac{1.4142-1}{1.4142+1}}}$

$=\sqrt{{\displaystyle \frac{0.4142}{2.4142}}}$

$=\sqrt{0.1715}$

$=0.4142$ 

17. $\sqrt[4]{\sqrt[3]{2^2}}$ equals

(A) $2^{{\displaystyle \frac{-1}{6}}}$            

(B)  $2^{-6}$           

(C)  $2^{{\displaystyle \frac{1}{6}}}$             

(D) $2^6$

Ans: C

Simplify it term by term.                          

$\sqrt[4]{\sqrt[3]{2^2}}$

$={\left({\left(2^2\right)}^{{\displaystyle \frac{1}{3}}}\right)}^{{\displaystyle \frac{1}{4}}}$

$=\left({{\left(2^2\right)}^{{\displaystyle \frac{1}{3}}\times }}^{{\displaystyle \frac{1}{4}}}\right)$

$={\left(2^2\right)}^{{\displaystyle \frac{1}{12}}}$

$={\left(2\right)}^{{\displaystyle \frac{2}{12}}}$

$={\left(2\right)}^{{\displaystyle \frac{1}{6}}}$ 

18. The product $\sqrt[3]{2}.\sqrt[4]{2}.\sqrt[12]{32}$ equals                       

(A) $\sqrt{2}$          

(B) $2$               

(C) $\sqrt[12]{2}$             

(D) $\sqrt[12]{32}$

Ans: B

$\sqrt[3]{2}\cdot \sqrt[4]{2}\cdot \sqrt[12]{32}$

$={\left(2\right)}^{{\displaystyle \frac{1}{3}}}\cdot {\left(2\right)}^{{\displaystyle \frac{1}{4}}}\cdot {\left(2\right)}^{{\displaystyle \frac{5}{12}}}$

$={\left(2\right)}^{{\displaystyle \frac{1}{3}}+{\displaystyle \frac{1}{4}}+{\displaystyle \frac{5}{2}}}$

$={\left(2\right)}^{{\displaystyle \frac{4+3+5}{12}}}$

$={\left(2\right)}^{{\displaystyle \frac{12}{12}}}$

$=2$ 

Hence, the value of $\sqrt[3]{2}\cdot \sqrt[4]{2}\cdot \sqrt[12]{32}$ is 2.

19. Value of $\sqrt[4]{{\left(81\right)}^{-2}}$ is 

(A) ${\displaystyle \frac{1}{9}}$      

(B) ${\displaystyle \frac{1}{3}}$             

(C) $9$             

(D) ${\displaystyle \frac{1}{81}}$

Ans: A

Let a be any integer $(a{}^m{)}^n=a^{mn}$

$$\sqrt[4]{{\left(81\right)}^{-2}}$$

$=\sqrt[4]{{\left({\displaystyle \frac{1}{81}}\right)}^2}$

$={\left({\left({\displaystyle \frac{1}{81}}\right)}^2\right)}^{{\displaystyle \frac{1}{4}}}$ 

$={\left({\displaystyle \frac{1}{81}}\right)}^{2\times {\displaystyle \frac{1}{4}}}$

$={\left({\displaystyle \frac{1}{81}}\right)}^{{\displaystyle \frac{1}{2}}}$

$={\displaystyle \frac{1}{9}}$ 

The value of $\sqrt[4]{{\left(81\right)}^{-2}}$ is $${\displaystyle \frac{1}{9}}$$ .

20. Value of $${\left(256\right)}^{0.16}\times {\left(256\right)}^{0.09}$$ is

(A) $$4$$             

(B) $$16$$              

(C) $$64$$               

(D) $$256.25$$

Ans: A

Let a and b are integers then $a^m.a^n=a^{m+n}$    

  ${\left(256\right)}^{0.16}\times {\left(256\right)}^{0.09}$

   $={\left(256\right)}^{0.16+0.09}$

   $={\left(256\right)}^{0.25}$

   $={\left(256\right)}^{{\displaystyle \frac{1}{4}}}$

   $={\left({\left(4\right)}^4\right)}^{{\displaystyle \frac{1}{4}}}$

   $={\left(4\right)}^{{\displaystyle \frac{4}{4}}}$

   $=4$ 

Hence, the value of $${\left(256\right)}^{0.16}\times {\left(256\right)}^{0.09}$$ is 4.

21. Which of the following is equal to$$x$$?

(A) $$x^{{\displaystyle \frac{12}{7}}}-x^{{\displaystyle \frac{5}{7}}}$$      

(B) $$\sqrt[12]{{\left(x^4\right)}^{{\displaystyle \frac{1}{3}}}}$$       

(C) $${\left(\sqrt{x^3}\right)}^{{\displaystyle \frac{2}{3}}}$$        

(D) $$x^{{\displaystyle \frac{12}{7}}}\times x^{{\displaystyle \frac{7}{12}}}$$

Ans: C

(A) $$x^{{\displaystyle \frac{12}{7}}}-x^{{\displaystyle \frac{5}{7}}}\ne x$$                

(B)                

$\sqrt[12]{{\left(x^4\right)}^{{\displaystyle \frac{1}{3}}}}$

$={\left(x\right)}^{4\times {\displaystyle \frac{1}{3}}\times {\displaystyle \frac{1}{12}}}$

$=x^{{\displaystyle \frac{1}{9}}}$

$\ne x$                   

(C)          

${\left(\sqrt{x^3}\right)}^{{\displaystyle \frac{2}{3}}}$

$={{\left(x^3\right)}^{{\displaystyle \frac{1}{2}}\times }}^{{\displaystyle \frac{2}{3}}}$

$=x^{{\displaystyle \frac{3}{3}}}$

$=x$                   

(D)  $$x^{{\displaystyle \frac{12}{7}}}\times x^{{\displaystyle \frac{7}{12}}}\ne x$$

So, option (C) is correct

Sample Questions

1. Are there two irrational numbers whose sum and product both are rational? Justify.

Ans: Yes, the sum and product of two irrational numbers are rational numbers.

Assume,

  $4+\sqrt{7}\mathrm{\&}4-\sqrt{7}\text{are two irrational numbers}$

  $(4+\sqrt{7})+(4-\sqrt{7})=8\text{ rational}$

  $(4+\sqrt{7})\times (4-\sqrt{7})=16-7=9\text{ rational}$ 

Hence we can conclude that two irrational numbers whose sum and product both are rational.

2. State whether the following statement is true: There is a number $$x$$ such that $$x^2$$ is irrational but $$x^4$$ is rational. Justify your answer by an example.

Ans: True, the Statement given is true there is a number $$x$$ such that $$x^2$$ is irrational but $$x^4$$ is rational.

Assume,

$$x=\sqrt[4]{2}$$ 

$$x^2=(\sqrt[4]{2}{)}^2=\sqrt{2}\text{ an irrational number}\text{.}$$ 

$$x^4=(\sqrt[4]{2}{)}^4=2,\text{ a rational number}\text{.}$$ 

Hence, the statement is “There is a number $$x$$ such that $$x^2$$ is irrational but $$x^4$$ is rational.”

EXERCISE 1.2

1. Let $$x$$ and $$y$$ be rational and irrational numbers, respectively. Is $$x+y$$ necessarily an irrational number? Give an example in support of your answer.

Ans: Yes, if $$x$$ and $$y$$ be rational and irrational numbers, respectively. Is $$x+y$$ necessarily an irrational number

Assume,  

  $x=5\mathrm{\&}y=\sqrt{7}$

  $x+y=5+\sqrt{7}(=p)\text{say}$ 

 $p^2={(5+\sqrt{7})}^2$ 

 $p^2=5^2+{(\sqrt{7})}^2+2(5)(\sqrt{7})$ 

 $p^2=25+7+10\sqrt{7}\Rightarrow {\displaystyle \frac{p^2-32}{10}}=\sqrt{7}$ 

$$x+y$$ is an irrational number.

2. Let $$x$$ be rational and $$y$$ be irrational. Is $$xy$$ necessarily irrational? Justify your answer by an example.

Ans: No, if $$x$$ be rational and $$y$$ be irrational then, $$xy$$ is not necessarily irrational

If we suppose $$x=2$$ be a non-zero rational and $y=\sqrt{5}$ be an irrational.

Then, $x\times y=2\times \sqrt{5}=2\sqrt{5}\text{ irrational}$ 

Hence, if $$x$$ be rational and $$y$$ be irrational then, $$xy$$ is not necessarily irrational

3. State whether the following statements are true or false? Justify your answer

(i). ${\displaystyle \frac{\sqrt{2}}{3}}$ is a rational number.

Ans: False, the division of an irrational number by a non-zero rational number gives an irrational number and here $\sqrt{2}$ is an irrational number and $3$ is a rational number.

(ii). There are infinitely many integers between any two integers.

Ans: False, As among two consecutive integers there is no existence of any integer.

(iii). Number of rational numbers between $$15$$  and $$18$$  is finite.

Ans: False, Among two rational numbers there exists an infinite number of rational numbers between them.

(iv). There are numbers which cannot be written in the form ${\displaystyle \frac{p}{q}}$ , $$q\ne 0,$$ $$p,q$$ both are integers.

Ans: True, As there exists an infinite numbers which cannot be written in the form of ${\displaystyle \frac{p}{q}}$ where $p$ and $q$ are integers and$q\ne 0$.

(v). The square of an irrational number is always rational.

Ans: False, it cannot be always true as for an irrational number ${\left(\sqrt[4]{3}\right)}^2=\sqrt{3}$ which is not a rational number.

(vi). $${\displaystyle \frac{\sqrt{12}}{\sqrt{3}}}$$ is not a rational number as $\sqrt{12}$ and $\sqrt{3}$ are not integers.

Ans: False, on solving $${\displaystyle \frac{\sqrt{12}}{\sqrt{3}}}$$ we get, 

$${\displaystyle \frac{\sqrt{12}}{\sqrt{3}}}={\displaystyle \frac{\sqrt{4\times 3}}{\sqrt{3}}}={\displaystyle \frac{\sqrt{4}\times \sqrt{3}}{\sqrt{3}}}=2\times 1=2$$ . Here, 2 is a rational number.

(vii). ${\displaystyle \frac{\sqrt{15}}{\sqrt{3}}}$ is written in the form ${\displaystyle \frac{p}{q}}$,  $$q\ne 0$$ and so it is a rational number.

Ans: False, on solving ${\displaystyle \frac{\sqrt{15}}{\sqrt{3}}}$ we get,

$${\displaystyle \frac{\sqrt{15}}{\sqrt{3}}}={\displaystyle \frac{\sqrt{5\times 3}}{\sqrt{3}}}={\displaystyle \frac{\sqrt{5}\times \sqrt{3}}{\sqrt{3}}}=\sqrt{5}$$. Here, $\sqrt{5}$ is an irrational number.

4. Classify the following numbers as rational or irrational with justification:

(i). $\sqrt{196}$ 

Ans: The given number is $\sqrt{196}$ 

Here, $\sqrt{196}=\sqrt{{\left(14\right)}^2}=14$ , rational number.

$\sqrt{196}$ is a rational number.

(ii). $3\sqrt{18}$ 

Ans: The given number is $3\sqrt{18}$

$$3\sqrt{18}=3\sqrt{{(3)}^2\times 2}=3\times 3\sqrt{2}=9\sqrt{2}$$

$$\sqrt{2}$$ is a non-repeating and non-terminating number. 

$$\Rightarrow 9\sqrt{2}$$ is a non-repeating and non-terminating number. 

$$\Rightarrow 3\sqrt{18}$$ is a non-repeating and non-terminating number. 

Hence, the number $$3\sqrt{18}$$cannot be represented as a ratio of two integers. 

Therefore, the number $$3\sqrt{18}$$ is an irrational number.

(iii). $\sqrt{{\displaystyle \frac{9}{27}}}$ 

Ans: The given number is$\sqrt{{\displaystyle \frac{9}{27}}}$.

$$\sqrt{{\displaystyle \frac{9}{27}}}=\sqrt{{\displaystyle \frac{9}{9\times 3}}}={\displaystyle \frac{1}{\sqrt{3}}}$$

 $$\sqrt{3}$$ is an irrational number. 

 $$\Rightarrow {\displaystyle \frac{1}{\sqrt{3}}}$$ is an irrational number.

 $\Rightarrow \sqrt{{\displaystyle \frac{9}{27}}}$is an irrational number.

(iv). $\sqrt{{\displaystyle \frac{28}{343}}}$ 

Ans: The given number is $\sqrt{{\displaystyle \frac{28}{343}}}$

$${\displaystyle \frac{\sqrt{28}}{\sqrt{343}}}={\displaystyle \frac{\sqrt{2\times 2\times 7}}{\sqrt{7\times 7\times 7}}}={\displaystyle \frac{2\sqrt{7}}{7\sqrt{7}}}={\displaystyle \frac{2}{7}}$$, simple fraction

And we know that rational numbers are the real numbers that can be written in simple fractions.

 The number $\sqrt{{\displaystyle \frac{28}{343}}}$ is a rational number.

(v). $-\sqrt{0.4}$ 

Ans: The given number is $-\sqrt{0.4}$

$$-\sqrt{0.4}=-\sqrt{{\displaystyle \frac{4}{10}}}=-{\displaystyle \frac{2}{\sqrt{10}}}$$

Here, $$\sqrt{10}$$ is an irrational number.

$$\Rightarrow -{\displaystyle \frac{2}{\sqrt{10}}}$$ is an irrational number.

$\Rightarrow -\sqrt{0.4}$is an irrational number.

(vi). ${\displaystyle \frac{\sqrt{12}}{\sqrt{75}}}$ 

Ans: The given number is ${\displaystyle \frac{\sqrt{12}}{\sqrt{75}}}$.

$${\displaystyle \frac{\sqrt{12}}{\sqrt{75}}}={\displaystyle \frac{\sqrt{4\times 3}}{\sqrt{25\times 3}}}={\displaystyle \frac{\sqrt{4}\sqrt{3}}{\sqrt{25}\sqrt{3}}}={\displaystyle \frac{2}{5}}$$, simple fraction

And we know that rational numbers are the real numbers that can be written in simple fractions.

The number ${\displaystyle \frac{\sqrt{12}}{\sqrt{75}}}$ is rational number.

(vii). $$0.5918$$ 

Ans: The given number is $$0.5918$$.

$$0.5918$$ is a terminating decimal so it can be written in the form of ${\displaystyle \frac{p}{q}}$ where $p$ and $q$ are integers and$q\ne 0$

Hence, the number 0.5918 is a rational number.

(viii). $$\left(1+\sqrt{5}\right)-\left(4+\sqrt{5}\right)$$ 

Ans: The given number is $$\left(1+\sqrt{5}\right)-\left(4+\sqrt{5}\right)$$

$$(1+\sqrt{5})-(4+\sqrt{5})=1-4+\sqrt{5}-\sqrt{5}=-3$$, rational number.                    

 Hence, $$\left(1+\sqrt{5}\right)-\left(4+\sqrt{5}\right)$$ is rational number.         

(ix). $$10.124124...$$ 

Ans: The given number is $$10.124124...$$ 

 When any rational number is expressed in the form of decimal expansion then it is either terminating or non-terminating recurring.

$$10.124124...$$ is a non-terminating recurring decimal expansion so is a rational number.

(x). $$1.010010001...$$ 

Ans: The given number is $$1.010010001...$$

$$1.010010001...$$ is a non-terminating non-recurring decimal expansion so is an irrational number.

$$1.010010001...$$ is an irrational number.

Sample Questions

1. Locate $\sqrt{13}$ on the number line.

Ans:

(i). Write 13 as a square of two natural numbers $$13=9+4=3^2+2^2$$ 

(ii). Draw $OA$ of measurement $3\text{units}$ on the number line and then $BA$ of $2\text{units}$ in perpendicular to $OA$.

(iii). Taking O as a centre and radius equal to $OB$, draw an arc which will intersect the number line at point P.

(iv). A right triangle with known leg lengths determines a hypotenuse that is of equal length to the number you want to place on the number line.

(v). Point P is refer to $\sqrt{13}$                                                     

2. Express $$0.12\bar{3}$$ in the form ${\displaystyle \frac{p}{q}}$ where $$p$$ and $$q$$ are integers and $$q\ne 0.$$ 

Ans: Assume, $$x=0.12\overline{3}$$

  $10x=1.2\bar{3}$

  $10x-x=1.2\bar{3}-0.12\bar{3}=1.233\bar{3}...-0.1233\bar{3}...$

  $9x=1.11$

  $x={\displaystyle \frac{1.11}{9}}$

  $x={\displaystyle \frac{111}{900}}$

  $x={\displaystyle \frac{37}{300}}$ 

 Hence, the number $$0.12\bar{3}$$ in the form ${\displaystyle \frac{p}{q}}$is $${\displaystyle \frac{37}{300}}$$ .

3. Simplify: $(3\sqrt{5}-5\sqrt{2})(4\sqrt{5}+3\sqrt{2})$ 

Ans: The given expression is 

$(3\sqrt{5}-5\sqrt{2})(4\sqrt{5}+3\sqrt{2})$

$\Rightarrow 12\times 5-20\sqrt{2}\times \sqrt{5}+9\sqrt{5}\times \sqrt{2}-15\times 2$

$\text{Add the like terms, we get:}$

$\Rightarrow 60-20\sqrt{10}+9\sqrt{10}-30$

$\Rightarrow 30-11\sqrt{10}$  

Hence, $(3\sqrt{5}-5\sqrt{2})(4\sqrt{5}+3\sqrt{2})=30-11\sqrt{10}$

4. Find the value of $$a$$ in the following:

${\displaystyle \frac{6}{3\sqrt{2}-2\sqrt{3}}}=3\sqrt{2}-a\sqrt{3}$ 

Ans: Given expression is ${\displaystyle \frac{6}{3\sqrt{2}-2\sqrt{3}}}=3\sqrt{2}-a\sqrt{3}$

Consider the LHS.

${\displaystyle \frac{6}{3\sqrt{2}-2\sqrt{3}}}$ 

Multiply the numerator and denominator by $3\sqrt{2}+2\sqrt{3}$ , we get:

$\Rightarrow {\displaystyle \frac{6}{3\sqrt{2}-2\sqrt{3}}}\times {\displaystyle \frac{3\sqrt{2}+2\sqrt{3}}{3\sqrt{2}+2\sqrt{3}}}$ 

On solving,

$\Rightarrow {\displaystyle \frac{6(3\sqrt{2}+2\sqrt{3})}{{(3\sqrt{2})}^2-{(2\sqrt{3})}^2}}$

$\Rightarrow {\displaystyle \frac{6(3\sqrt{2}+2\sqrt{3})}{18-12}}$

$\Rightarrow {\displaystyle \frac{6(3\sqrt{2}+2\sqrt{3})}{6}}=3\sqrt{2}+2\sqrt{3}$ 

On equating L.H.S and R.H.S

$3\sqrt{2}+2\sqrt{3}=3\sqrt{2}-a\sqrt{3}$

  $\text{Comparing the like terms, we get:}$

  $a=-2$ 

5. Simplify: ${\left[5{\left(8^{{\displaystyle \frac{1}{3}}}+27^{{\displaystyle \frac{1}{3}}}\right)}^3\right]}^{{\displaystyle \frac{1}{4}}}$  

Ans: ${\left(a^p\right)}^q={\left(a\right)}^{pq}$ 

${\left[5{\left(8^{{\displaystyle \frac{1}{3}}}+27^{{\displaystyle \frac{1}{3}}}\right)}^3\right]}^{{\displaystyle \frac{1}{4}}}$

$={\left[5{\left({\left(2^3\right)}^{{\displaystyle \frac{1}{3}}}+{\left(3^3\right)}^{{\displaystyle \frac{1}{3}}}\right)}^3\right]}^{{\displaystyle \frac{1}{4}}}$ 

$={\left[5{(2+3)}^3\right]}^{{\displaystyle \frac{1}{4}}}$

$={\left[5{(5)}^3\right]}^{{\displaystyle \frac{1}{4}}}$

$={\left[5^4\right]}^{{\displaystyle \frac{1}{4}}}=5$ 

EXERCISE 1.3

1. Find which of the variables $$x$$,$$y$$,$$z$$ and $$u$$ represent rational numbers and which irrational numbers:

(i) $x^2=5$         

Ans: $x^2=5$

$\therefore x=\pm \sqrt{5},\text{ irrational number}$ 

So, $$x$$ represents irrational number

(ii)$y^2=9$      

Ans:  $y^2=9$

$\therefore y=\pm 3,\text{ rational number}$
So,$$y$$ represents rational number

(iii) $$z^2=.04$$

Ans: $z^2=.04$

  $z=\pm 0.2,\text{ rational number}$ 

So, $$z$$ represents rational number

(iv) $$u^2={\displaystyle \frac{17}{4}}$$

Ans: $u^2={\displaystyle \frac{17}{4}}$

$\therefore u=\pm \sqrt{{\displaystyle \frac{17}{4}}}$

$\therefore u=\pm {\displaystyle \frac{\sqrt{17}}{2}},\text{ irrational number}$

So, $$u$$ represents irrational number

2. Find three rational numbers between

(i) $$-1$$ and $$2$$       

Ans: Rational numbers can be easily found between $$-1$$ and $$\text{2}$$  and  those are $${\displaystyle \frac{1}{2}}$$,$${\displaystyle \frac{2}{4}}$$ and $${\displaystyle \frac{4}{8}}$$

(ii) $$0.1$$ and $$0.11$$   

Ans: $$0.1={\displaystyle \frac{1}{10}}$$ and $$0.11={\displaystyle \frac{11}{100}}$$

So, $${\displaystyle \frac{1}{10}}\times {\displaystyle \frac{100}{100}}={\displaystyle \frac{100}{1000}}$$ and $${\displaystyle \frac{11}{100}}\times {\displaystyle \frac{10}{10}}={\displaystyle \frac{110}{1000}}$$

Hence, three rational numbers between $${\displaystyle \frac{100}{1000}}$$  and  $${\displaystyle \frac{110}{1000}}$$ are $${\displaystyle \frac{102}{1000}}$$,$${\displaystyle \frac{104}{1000}}$$ and $${\displaystyle \frac{106}{1000}}$$.

(iii) $${\displaystyle \frac{5}{7}}$$ and $${\displaystyle \frac{6}{7}}$$   

Ans: $${\displaystyle \frac{5}{7}}\times {\displaystyle \frac{10}{10}}={\displaystyle \frac{50}{70}}$$ and $${\displaystyle \frac{6}{7}}\times {\displaystyle \frac{10}{10}}={\displaystyle \frac{60}{70}}$$

Therefore, three rational numbers between $${\displaystyle \frac{50}{70}}$$ and $${\displaystyle \frac{60}{70}}$$ are $${\displaystyle \frac{51}{70}}$$, $${\displaystyle \frac{52}{70}}$$ and $${\displaystyle \frac{53}{70}}$$

(iv) $${\displaystyle \frac{1}{4}}$$ and $${\displaystyle \frac{1}{5}}$$

Ans: $${\displaystyle \frac{1}{4}}\times {\displaystyle \frac{50}{50}}={\displaystyle \frac{50}{200}}$$ and $${\displaystyle \frac{1}{5}}\times {\displaystyle \frac{40}{40}}={\displaystyle \frac{40}{200}}$$

So, three rational numbers between $${\displaystyle \frac{50}{200}}$$ and $${\displaystyle \frac{40}{200}}$$ are $${\displaystyle \frac{41}{200}}$$, $${\displaystyle \frac{42}{200}}$$ and $${\displaystyle \frac{43}{200}}$$ 

3. Insert a rational number and an irrational number between the following:

(i). $$2$$ and $$3$$    

Ans: The given numbers are 2 and 3.

 Rational number between $$2$$ and $$3$$ is 2.1

 Irrational number between $$2$$ and $$3$$ is 2.050050005……

(ii) $$0$$ and $$0.1$$   

Ans: The given numbers are 0 and 0.1.

Rational number between 0 and 1is 0.05

Irrational number between 0 and 1 is 0.003000300003……

(iii) $${\displaystyle \frac{1}{3}}$$ and $${\displaystyle \frac{1}{2}}$$ 

Ans: The given numbers are $${\displaystyle \frac{1}{3}}$$ and $${\displaystyle \frac{1}{2}}$$.

Rational number between $${\displaystyle \frac{1}{3}}$$ and $${\displaystyle \frac{1}{2}}$$ is $${\displaystyle \frac{5}{12}}$$.

Irrational number between $${\displaystyle \frac{1}{3}}$$ and $${\displaystyle \frac{1}{2}}$$ is 0.4242242224……..