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# R.D. Sharma solutions for Class 9 Mathematics chapter 3 - Rationalisation

## Chapter 3 - Rationalisation

#### Pages 0 - 3

Is zero a rational number? Can you write it in the form p/q, where p and q are integersand q ≠ 0?

Simplify of the following:

root(3)4  xx root(3)16

Simplify the following

3(a^4b^3)^10xx5(a^2b^2)^3

Simplify of the following:

root(4)1250/root(4)2

Simplify the following

(2x^-2y^3)^3

Simplify the following

((4xx10^7)(6xx10^-5))/(8xx10^4)

Simplify the following

(4ab^2(-5ab^3))/(10a^2b^2)

Simplify the following

((x^2y^2)/(a^2b^3))^n

Simplify the following

(a^(3n-9))^6/(a^(2n-4))

Find five rational numbers between 1 and 2.

If a = 3 and b = -2, find the values of :

aa + bb

Simplify the following expressions:

(4 + sqrt7)(3 + sqrt2)

Simplify the following expressions:

(3 + sqrt3)(5 - sqrt2)

If a = 3 and b = -2, find the values of :

ab + ba

Simplify the following expressions:

(sqrt5 - 2)(sqrt3 - sqrt5)

If a = 3 and b = -2, find the values of :

(a + b)ab

Find six rational numbers between 3 and 4.

Simplify the following expressions:

(11 + sqrt11)(11 - sqrt11)

Prove that:

(x^a/x^b)^(a^2+ab+b^2)xx(x^b/x^c)^(b^2+bc+c^2)xx(x^c/x^a)^(c^2+ca+a^2)=1

Simplify the following expressions:

(5 + sqrt7)(5 - sqrt7)

Prove that:

(x^a/x^b)^cxx(x^b/x^c)^axx(x^c/x^a)^b=1

Simplify the following expressions:

(sqrt8 - sqrt2)(sqrt8 + sqrt2)

Simplify the following expressions:

(3 + sqrt3)(3 - sqrt3)

Simplify the following expressions:

(sqrt5 - sqrt2)(sqrt5 + sqrt2)

Find five rational numbers between 3/5 and 4/5.

Prove that:

1/(1+x^(a-b))+1/(1+x^(b-a))=1

Simplify the following expressions:

(sqrt3 + sqrt7)^2

Prove that:

1/(1+x^(b-a)+x^(c-a))+1/(1+x^(a-b)+x^(c-b))+1/(1+x^(b-c)+x^(a-c))=1

Simplify the following expressions:

(sqrt5 - sqrt3)^2

Simplify the following expressions:

(2sqrt5 + 3sqrt2)^2

Prove that:

(a+b+c)/(a^-1b^-1+b^-1c^-1+c^-1a^-1)=abc

State whether the following statement is true or false. Give reasons for your answers.

Every whole number is a natural number.

State whether the following statement is true or false. Give reasons for your answers.

Every integer is a rational number.

Prove that:

(a^-1+b^-1)^-1=(ab)/(a+b)

State whether the following statement is true or false. Give reasons for your answers.

Every rational number is an integer.

State whether the following statement is true or false. Give reasons for your answers.

Every natural number is a whole number.

State whether the following statement is true or false. Give reasons for your answers.

Every integer is a whole number.

State whether the following statement is true or false. Give reasons for your answers.

Every rational number is a whole number.

If abc = 1, show that 1/(1+a+b^-1)+1/(1+b+c^-1)+1/(1+c+a^-1)=1

Simplify the following:

(3^nxx9^(n+1))/(3^(n-1)xx9^(n-1))

Simplify the following:

(5xx25^(n+1)-25xx5^(2n))/(5xx5^(2n+3)-25^(n+1))

Simplify the following:

(5^(n+3)-6xx5^(n+1))/(9xx5^x-2^2xx5^n)

Simplify the following:

(6(8)^(n+1)+16(2)^(3n-2))/(10(2)^(3n+1)-7(8)^n)

Solve the following equation for x:

7^(2x+3)=1

Solve the following equation for x:

2^(x+1)=4^(x-3)

Solve the following equation for x:

2^(5x+3)=8^(x+3)

Solve the following equation for x:

4^(2x)=1/32

Solve the following equation for x:

4^(x-1)xx(0.5)^(3-2x)=(1/8)^x

Solve the following equation for x:

2^(3x-7)=256

Solve the following equations for x:

2^(2x)-2^(x+3)+2^4=0

Solve the following equations for x:

3^(2x+4)+1=2.3^(x+2)

If 49392 = a4b2c3, find the values of a, b and c, where a, b and c are different positive primes.

If 1176=2^a3^b7^c, find a, b and c.

Given 4725=3^a5^b7^c, find

(i) the integral values of a, b and c

(ii) the value of 2^-a3^b7^c

If a=xy^(p-1), b=xy^(q-1) and c=xy^(r-1), prove that a^(q-r)b^(r-p)c^(p-q)=1

#### Pages 0 - 15

Assuming that x, y, z are positive real numbers, simplify the following:

(sqrt(x^-3))^5

Express the following rational number as decimal:

42/100

Rationalise the denominator of each of the following

3/sqrt5

Assuming that x, y, z are positive real numbers, simplify the following:

sqrt(x^3y^-2)

Express the following rational number as decimal:

327/500

Rationalise the denominator of each of the following

3/(2sqrt5)

Assuming that x, y, z are positive real numbers, simplify the following:

(x^((-2)/3)y^((-1)/2))^2

Express the following rational number as decimal:

15/4

Rationalise the denominator of each of the following

1/sqrt12

Assuming that x, y, z are positive real numbers, simplify the following:

(sqrtx)^((-2)/3)sqrt(y^4)divsqrt(xy^((-1)/2))

Rationalise the denominator of the following

sqrt2/sqrt5

Rationalise the denominator of each of the following

(sqrt3 + 1)/sqrt2

Assuming that x, y, z are positive real numbers, simplify the following:

root5(243x^10y^5z^10)

Rationalise the denominator of the following

(sqrt2 + sqrt5)/3

Assuming that x, y, z are positive real numbers, simplify the following:

(x^-4/y^-10)^(5/4)

Assuming that x, y, z are positive real numbers, simplify the following:

(sqrt2/sqrt3)^5(6/7)^2

Rationalise the denominator of the following

(3sqrt2)/sqrt5

Express the following rational number as decimal:

2/3

Simplify:

(16^(-1/5))^(5/2)

Find the value to three places of decimals of the following. It is given that

sqrt2 = 1.414, sqrt3 = 1.732, sqrt5 = 2.236 and sqrt10 = 3.162

2/sqrt3

Express the following rational number as decimal:

-4/9

Find the value to three places of decimals of the following. It is given that

sqrt2 = 1.414, sqrt3 = 1.732, sqrt5 = 2.236 and sqrt10 = 3.162

3/sqrt10

Simplify:

root5((32)^-3)

Simplify:

root3((343)^-2)

Express the following rational number as decimal:

-2/15

Find the value to three places of decimals of the following. It is given that

sqrt2 = 1.414, sqrt3 = 1.732, sqrt5 = 2.236 and sqrt10 = 3.162

(sqrt5 + 1)/sqrt2

Express the following rational number as decimal:

-22/13

Simplify:

(0.001)^(1/3)

Find the value to three places of decimals of the following. It is given that

sqrt2 = 1.414, sqrt3 = 1.732, sqrt5 = 2.236 and sqrt10 = 3.162

(sqrt10 + sqrt15)/sqrt2



Simplify:

((25)^(3/2)xx(243)^(3/5))/((16)^(5/4)xx(8)^(4/3))

Find the value to three places of decimals of the following. It is given that

sqrt2 = 1.414, sqrt3 = 1.732, sqrt5 = 2.236 and sqrt10 = 3.162

(2 + sqrt3)/3

Express the following rational number as decimal:

437/999

Find the value to three places of decimals of the following. It is given that

sqrt2 = 1.414, sqrt3 = 1.732, sqrt5 = 2.236 and sqrt10 = 3.162

(sqrt2 - 1)/sqrt5

Express the following rational number as decimal:

33/26

Simplify:

(sqrt2/5)^8div(sqrt2/5)^13

Simplify:

((5^-1xx7^2)/(5^2xx7^-4))^(7/2)xx((5^-2xx7^3)/(5^3xx7^-5))^(-5/2)

Look at several examples of rational numbers in the form p/q (q≠0), where p and q are integers with no common factors other than 1 and having terminating decimal representations (expansions). Can you guess what property q must satisfy?

Prove that:

sqrt(3xx5^-3)divroot3(3^-1)sqrt5xxroot6(3xx5^6)=3/5

Express the following with rational denominator:

1/(3 + sqrt2)

Prove that:

9^(3/2)-3xx5^0-(1/81)^(-1/2)=15

Express of the following with rational denominator:

1/(sqrt6 - sqrt5)

Prove that:

(1/4)^-2-3xx8^(2/3)xx4^0+(9/16)^(-1/2)=16/3

Express the following with rational denominator:

16/(sqrt41 - 5)

Prove that:

(2^(1/2)xx3^(1/3)xx4^(1/4))/(10^(-1/5)xx5^(3/5))div(3^(4/3)xx5^(-7/5))/(4^(-3/5)xx6)=10

Express the following with rational denominator:

30/(5sqrt3 - 3sqrt5)

Prove that:

sqrt(1/4)+(0.01)^(-1/2)-(27)^(2/3)=3/2

Express the following with rational denominator:

1/(2sqrt5 - sqrt3)

Prove that:

(2^n+2^(n-1))/(2^(n+1)-2^n)=3/2

Express the following with rational denominator:

(sqrt3 + 1)/(2sqrt2 - sqrt3)

Express the following with rational denominator:

(6 - 4sqrt2)/(6 + 4sqrt2)

Prove that:

(64/125)^(-2/3)+1/(256/625)^(1/4)+(sqrt25/root3 64)=65/16

Express the following with rational denominator:

(3sqrt2 + 1)/(2sqrt5 - 3)

Prove that:

(3^-3xx6^2xxsqrt98)/(5^2xxroot3(1/25)xx(15)^(-4/3)xx3^(1/3))=28sqrt2

Express each one of the following with rational denominator:

(b^2)/(sqrt(a^2 + b^2) + a)

Prove that:

((0.6)^0-(0.1)^-1)/((3/8)^-1(3/2)^3+((-1)/3)^-1)=(-3)/2

Rationales the denominator and simplify:

(3 - sqrt2)/(3 + sqrt2)

Show that:

1/(1+x^(a-b))+1/(1+x^(b-a))=1

Show that:

[{x^(a(a-b))/x^(a(a+b))}div{x^(b(b-a))/x^(b(b+a))}]^(a+b)=1

Rationales the denominator and simplify:

(5 + 2sqrt3)/(7 + 4sqrt3)

Show that:

(x^(1/(a-b)))^(1/(a-c))(x^(1/(b-c)))^(1/(b-a))(x^(1/(c-a)))^(1/(c-b))=1

Rationales the denominator and simplify:

(1 + sqrt2)/(3 - 2sqrt2)

Show that:

(x^(a^2+b^2)/x^(ab))^(a+b)(x^(b^2+c^2)/x^(bc))^(b+c)(x^(c^2+a^2)/x^(ac))^(a+c)=x^(2(a^3+b^3+c^3))

Rationales the denominator and simplify:

(4sqrt3 + 5sqrt2)/(sqrt48 + sqrt18)

Show that:

(x^(a-b))^(a+b)(x^(b-c))^(b+c)(x^(c-a))^(c+a)=1

Rationales the denominator and simplify:

(2sqrt6 - sqrt5)/(3sqrt5 - 2sqrt6)

Show that:

{(x^(a-a^-1))^(1/(a-1))}^(a/(a+1))=x

Rationales the denominator and simplify:

(2sqrt3 - sqrt5)/(2sqrt2 + 3sqrt3)

Show that:

(a^(x+1)/a^(y+1))^(x+y)(a^(y+2)/a^(z+2))^(y+z)(a^(z+3)/a^(x+3))^(z+x)=1

Show that:

(3^a/3^b)^(a+b)(3^b/3^c)^(b+c)(3^c/3^a)^(c+a)=1

If 2x = 3y = 12z, show that 1/z=1/y+2/x

Simplify:

(5 + sqrt3)/(5 - sqrt3) + (5 - sqrt3)/(5 + sqrt3)

Simplify

1/(2 + sqrt3) + 2/(sqrt5 - sqrt3) + 1/(2 - sqrt5)

Simplify

2/(sqrt5 + sqrt3) + 1/(sqrt3 + sqrt2) + 3/(sqrt5 + sqrt2)

If 2x = 3y = 6-z, show that 1/x+1/y+1/z=0

In the following determine rational numbers a and b:

(sqrt3 - 1)/(sqrt3 + 1) = a - bsqrt3

In the following determine rational numbers a and b:

(4 + sqrt2)/(2 + sqrt2) = n - sqrtb

In the following determine rational numbers a and b:

(3 + sqrt2)/(3 - sqrt2) = a + bsqrt2

In the following determine rational numbers a and b:

(5 + 3sqrt3)/(7 + 4sqrt3) = a + bsqrt3

In the following determine rational numbers a and b:

(sqrt11 - sqrt7)/(sqrt11 + sqrt7) = a - bsqrt77

In the following determine rational numbers a and b:

(4 + 3sqrt5)/(4 - 3sqrt5) = a + bsqrt5

If ax = by = cz and b2 = ac, show that y=(2zx)/(z+x)

Find the value of 6/(sqrt5 - sqrt3) it being given that sqrt3 = 1.732 and  sqrt5 = 2.236

If 3x = 5y = (75)z, show that z=(xy)/(2x+y)

Find the values the following correct to three places of decimals, it being given that sqrt2 = 1.4142, sqrt3 = 1.732, sqrt5 = 2.2360, sqrt6 = 2.4495 and sqrt10 = 3.162

(3 - sqrt5)/(3 + 2sqrt5)

Find the values the following correct to three places of decimals, it being given that sqrt2 = 1.4142, sqrt3 = 1.732, sqrt5 = 2.2360, sqrt6 = 2.4495 and sqrt10 = 3.162

(1 + sqrt2)/(3 - 2sqrt2)

Simplify (3sqrt2 - 2sqrt3)/(3sqrt2 + 2sqrt3) + sqrt12/(sqrt3 - sqrt2)

If 27^x=9/3^x, find x.

if x = 2 +  sqrt3,find the value of x^2 + 1/x^2

Find the value of x in the following:

2^(5x)div2x=root5(2^20)

Find the value of x in the following:

(2^3)^4=(2^2)^x

Find the value of x in the following:

(3/5)^x(5/3)^(2x)=125/27

Find the value of x in the following:

5^(x-2)xx3^(2x-3)=135

Find the value of x in the following:

2^(x-7)xx5^(x-4)=1250

Find the value of x in the following:

(root3 4)^(2x+1/2)=1/32

Find the value of x in the following:

5^(2x+3)=1

Find the value of x in the following:

(13)^(sqrtx)=4^4-3^4-6

Find the value of x in the following:

(sqrt(3/5))^(x+1)=125/27

If x=2^(1/3)+2^(2/3), Show that x3 - 6x = 6

if   x= 3 + sqrt8, find the value of x^2 + 1/x^2

if x =  (sqrt3 + 1)/2 find the value of 4x^2 +2x^2 - 8x + 7

Determine (8x)^x,If 9^(x+2)=240+9^x

If 3^(x+1)=9^(x-2), find the value of 2^(1+x)

If 3^(4x) = (81)^-1 and 10^(1/y)=0.0001, find the value of 2^(-x+4y).

If 5^(3x)=125 and 10^y=0.001, find x and y.

Solve the following equation:

3^(x+1)=27xx3^4

Solve the following equation:

4^(2x)=(root3 16)^(-6/y)=(sqrt8)^2

Solve the following equation:

3^(x-1)xx5^(2y-3)=225

Solve the following equation:

8^(x+1)=16^(y+2) and, (1/2)^(3+x)=(1/4)^(3y)

Solve the following equation:

4^(x-1)xx(0.5)^(3-2x)=(1/8)^x

Solve the following equation:

sqrt(a/b)=(b/a)^(1-2x), where a and b are distinct primes.

If a and b are distinct primes such that root3 (a^6b^-4)=a^xb^(2y), find x and y.

If a and b are different positive primes such that

((a^-1b^2)/(a^2b^-4))^7div((a^3b^-5)/(a^-2b^3))=a^xb^y, find x and y.

If a and b are different positive primes such that

(a+b)^-1(a^-1+b^-1)=a^xb^y, find x + y + 2.

If 2^x xx3^yxx5^z=2160, find x, y and z. Hence, compute the value of 3^x xx2^-yxx5^-z.

If 1176 = 2^axx3^bxx7^c, find the values of a, b and c. Hence, compute the value of 2^axx3^bxx7^-c as a fraction.

Simplify:

(x^(a+b)/x^c)^(a-b)(x^(b+c)/x^a)^(b-c)(x^(c+a)/x^b)^(c-a)

Simplify:

root(lm)(x^l/x^m)xxroot(mn)(x^m/x^n)xxroot(nl)(x^n/x^l)

Show that:

((a+1/b)^mxx(a-1/b)^n)/((b+1/a)^mxx(b-1/a)^n)=(a/b)^(m+n)

If a=x^(m+n)y^l, b=x^(n+l)y^m and c=x^(l+m)y^n, Prove that a^(m-n)b^(n-l)c^(l-m)=1

If x = a^(m+n), y=a^(n+l) and z=a^(l+m), prove that x^my^nz^l=x^ny^lz^m

#### Page 0

Express the following decimal in the form p/q : 0.39

Express the following decimal in the form p/q : 0.750

Express the following decimal in the form p/q : 2.15

Express the following decimal in the form p/q:

7.010

Express the following decimal in the form p/q: 9.90

Express the following decimal in the form p/q: 1.0001

Express the following decimal in the form p/q: 0.bar4

Express the following decimal in the form p/q: 0.bar37

Express the following decimal in the form p/q: 0.bar54

Express the following decimal in the form p/q: 0.bar621

Express the following decimal in the form p/q: 125.bar3

Express the following decimal in the form p/q: 4.bar7

Express the following decimal in the form p/q: 0.4bar7

#### Page 0

Define an irrational number ?

Explain, how irrational numbers differ from rational numbers ?

Examine, whether the following number are rational or irrational:

sqrt7

Examine, whether the following number are rational or irrational:

sqrt4

Examine, whether the following number are rational or irrational:

2+sqrt3

Examine, whether the following number are rational or irrational:

sqrt3+sqrt2

Examine, whether the following number are rational or irrational:

sqrt3+sqrt5

Examine, whether the following number are rational or irrational:

(sqrt2-2)^2

Examine, whether the following number are rational or irrational:

(2-sqrt2)(2+sqrt2)

Examine, whether the following number are rational or irrational:

(sqrt2+sqrt3)^2

Examine, whether the following number are rational or irrational:

sqrt5-2

Classify the following number as rational or irrational :-

sqrt23

Classify the following number as rational or irrational :-

sqrt225

Classify the following number as rational or irrational :-

0.3796

Classify the following number as rational or irrational :-

7.478478...

Classify the following number as rational or irrational :-

1.1010010001...

Identify the following as rational or irrational number. Give the decimal representation of rational number:

sqrt4

Identify the following as rational or irrational number. Give the decimal representation of rational number:

3sqrt18

Identify the following as rational or irrational number. Give the decimal representation of rational number:

sqrt1.44

Identify the following as rational or irrational number. Give the decimal representation of rational number:

sqrt(9/27)

Identify the following as rational or irrational number. Give the decimal representation of rational number:

-sqrt64

Identify the following as rational or irrational number. Give the decimal representation of rational number:

sqrt100

In the following equation, find which variables x, y, z etc. represent rational or irrational number:

x2 = 5

In the following equation, find which variables x, y, z etc. represent rational or irrational number:

y2 = 9

In the following equation, find which variables x, y, z etc. represent rational or irrational number:

z2 = 0.04

In the following equation, find which variables x, y, z etc. represent rational or irrational number:

u^2=17/4

In the following equation, find which variables x, y, z etc. represent rational or irrational number:

v2 = 3

In the following equation, find which variables x, y, z etc. represent rational or irrational number:

w2 = 27

In the following equation, find which variables x, y, z etc. represent rational or irrational number:

t2 = 0.4

Give two rational numbers lying between 0.232332333233332... and 0.212112111211112.

Give two rational numbers lying between 0.515115111511115... and 0.535335333533335...

Find one irrational number between 0.2101 and 0.222... = 0.bar2

Find a rational number and also an irrational number lying between the numbers 0.3030030003... and 0.3010010001...

Find three different irrational numbers between the rational numbers 5/7" and "9/11.

Give an example of two irrational numbers whose:

difference is a rational number.

Give an example of two irrational numbers whose:

difference is an irrational number.

Give an example of two irrational numbers whose:

sum is a rational number.

Give an example of two irrational numbers whose:

sum is an irrational number.

Give an example of two irrational numbers whose:

product is an rational number.

Give an example of two irrational numbers whose:

product is an irrational number.

Give an example of two irrational numbers whose:

quotient is a rational number.

Give an example of two irrational numbers whose:

quotient is an irrational number.

Find two irrational numbers between 0.5 and 0.55.

Find two irrational numbers lying between 0.1 and 0.12.

Prove that sqrt3+sqrt5 is an irrational number.

#### Page 0

Complete the following sentence:

Every point on the number line corresponds to a _________ number which many be either _______ or ________.

Complete the following sentence:

The decimal form of an irrational number is neither ________ nor _________

Complete the following sentence:

The decimal representation of a rational number is either ______ or _________.

Complete the following sentence:

Every real number is either ______ number or _______ number.

Find whether the following statement is true or false.

Every real number is either rational or irrational.

Find whether the following statement is true or false.

π is an irrational number.

Find whether the following statement is true or false.

Irrational numbers cannot be represented by points on the number line.

Represent sqrt6, sqrt7, sqrt8 on the number line.

Represent sqrt3.5, sqrt9.4, sqrt10.5 on the real number line.

#### Page 0

Visualise 2.665 on the number line, using successive magnification.

Visualise the representation of 5.3bar7` on the number line upto 5 decimal places, that is upto 5.37777.

## R.D. Sharma solutions for Class 9 Mathematics chapter 3 - Rationalisation

R.D. Sharma solutions for Class 9 Mathematics chapter 3 (Rationalisation) include all questions with solution and detail explanation from Mathematics for Class 9 by R D Sharma (2018-19 Session). This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has created the CBSE Mathematics for Class 9 by R D Sharma (2018-19 Session) solutions in a manner that help students grasp basic concepts better and faster.

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Concepts covered in Class 9 Mathematics chapter 3 Rationalisation are Introduction of Real Number, Irrational Numbers, Real Numbers and Their Decimal Expansions, Representing Real Numbers on the Number Line, Operations on Real Numbers, Laws of Exponents for Real Numbers.

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