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RD Sharma Solution for Class 9 Mathematics Chapter 3 - Rationalisation

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Our RD Sharma Textbook Solutions are considered extremely helpful for solving the tough questions which are asked in the CBSE Class 9 exam. TopperLearning Textbook Solutions are compiled by subject experts. Herein, you can find all the answers to the textbook questions for Chapter 3 - Rationalisation.

All our solutions are created in accordance with the latest CBSE syllabus, and they are amended from time to time to provide the most relevant answers. Our free RD Sharma Solutions for CBSE Class 9 Mathematics will strengthen your fundamentals in Mathematics and will help you in your attempts to score more marks in the final examination. CBSE Class 9 students can refer to our solutions any time — while doing their homework and while preparing for the exam.

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Exercise/Page

RD Sharma Solution for Class 9 Mathematics Chapter 3 - Rationalisation Page/Excercise 3.1

Solution 1

Solution 2

Solution 3


Solution 4


RD Sharma Solution for Class 9 Mathematics Chapter 3 - Rationalisation Page/Excercise 3.2

Solution 1(i)

Solution 1(ii)

Solution 1(iii)


Solution 1(iv)

Solution 1(v)

Solution 1(vi)

Solution 1(vii)

Solution 2(i)

Solution 2(ii)

Solution 2(iii)

Solution 2(iv)

Solution 2(v)

Solution 2(vi)

Solution 3(i)

Solution 3(ii)

Solution 3(iii)

Solution 3(iv)

Solution 3(v)

Solution 3(vi)

Solution 3(vii)

Solution 3(viii)

Solution 3(ix)

Solution 4(i)

Solution 4(ii)

Solution 4(iii)

Solution 4(iv)

Solution 4(v)

Solution 4(vi)

Solution 5(i)

Solution 5(ii)

Solution 5(iii)

Solution 6(i)

Solution 6(ii)


Solution 6(iii)


Solution 6(iv)


Solution 6(v)


Solution 6(vi)


Solution 7

Solution 8(i)

Solution 8(ii)

Solution 10

Solution 11

Solution 12

Solution 9(i)

  

Solution 9(ii)

  

RD Sharma Solution for Class 9 Mathematics Chapter 3 - Rationalisation Page/Excercise 3.16

Solution 1

begin mathsize 11px style 10 space equals space 5 space straight x space 2
15 space equals space 5 space straight x space 3
therefore space square root of 10 space cross times space square root of 15 space equals space square root of 5 cross times 2 end root space cross times space square root of 5 cross times 3 end root space
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space square root of 5 cross times square root of 2 cross times square root of 5 cross times square root of 3
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space open parentheses square root of 5 cross times square root of 5 close parentheses space cross times space square root of 2 cross times square root of 3
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space 5 square root of 6
Hence comma space straight c orrect space option space is space left parenthesis straight a right parenthesis. end style

Solution 2

begin mathsize 11px style fifth root of 6 space space equals space open parentheses 6 close parentheses to the power of 1 divided by 5 end exponent
so space fifth root of 6 space cross times fifth root of 6 space space equals space open parentheses 6 close parentheses to the power of 1 divided by 5 end exponent space cross times space open parentheses 6 close parentheses to the power of 1 divided by 5 end exponent
space space space space space space space space space space space space space space space space space space space space space space space space space space equals open parentheses 6 cross times 6 close parentheses to the power of 1 divided by 5 end exponent
space space space space space space space space space space space space space space space space space space space space space space space space space space space equals open parentheses 36 close parentheses to the power of 1 divided by 5 end exponent
space space space space space space space space space space space space space space space space space space space space space space space space space space space equals fifth root of 36
Hence comma space correct space option space is space open parentheses straight a close parentheses. end style

RD Sharma Solution for Class 9 Mathematics Chapter 3 - Rationalisation Page/Excercise 3.17

Solution 3

begin mathsize 11px style Rationalisation space factor space of space any space number space like space space square root of straight a space is space fraction numerator 1 over denominator square root of straight a end fraction space or space fraction numerator 1 over denominator square root of straight a end fraction space is space square root of straight a. space space
So comma space Rationalisation space factor space of space square root of 3 space is space fraction numerator 1 over denominator square root of 3 end fraction.
Hence comma space correct space option space is space left parenthesis straight b right parenthesis. end style

Solution 4

begin mathsize 11px style Rationalisation space factor space of space any space number space straight a plus-or-minus square root of straight b space is space straight a space minus-or-plus square root of straight b.
So comma space Rationalisation space factor space of space 2 plus square root of 3 space will space be space 2 space minus space square root of 3
Hence comma space correct space option space is space left parenthesis straight a right parenthesis. end style

Solution 5

begin mathsize 11px style straight x equals square root of 5 plus 2
rightwards double arrow 1 over straight x equals fraction numerator 1 over denominator square root of 5 plus 2 end fraction equals fraction numerator 1 over denominator square root of 5 plus 2 end fraction straight space fraction numerator square root of 5 minus 2 over denominator square root of 5 space minus space 2 end fraction equals straight space fraction numerator square root of 5 minus 2 over denominator 5 minus 4 end fraction space equals space fraction numerator square root of 5 minus 2 over denominator 1 end fraction equals square root of 5 minus 2
Now comma space space straight x minus 1 over straight x straight space equals square root of 5 plus 2 minus open parentheses square root of 5 minus 2 close parentheses equals square root of 5 plus 2 minus square root of 5 plus 2 equals 4
Hence comma space correct space option space is space left parenthesis straight b right parenthesis. end style

Solution 6

fraction numerator square root of 3 minus 1 over denominator square root of 3 plus 1 end fraction
Multiplying space and space dividing space by space the space rationalisation space factor
of space denominator comma space we space get
fraction numerator square root of 3 minus 1 over denominator square root of 3 plus 1 end fraction cross times fraction numerator square root of 3 minus 1 over denominator square root of 3 minus 1 end fraction
equals fraction numerator open parentheses square root of 3 minus 1 close parentheses squared over denominator open parentheses square root of 3 close parentheses squared minus 1 squared end fraction
equals fraction numerator 3 minus 2 square root of 3 plus 1 over denominator 3 minus 1 end fraction
equals fraction numerator 4 minus 2 square root of 3 over denominator 2 end fraction
equals fraction numerator 2 open parentheses 2 minus square root of 3 close parentheses over denominator 2 end fraction
equals 2 minus square root of 3
Comparing space with space straight a minus straight b square root of 3 comma space we space get space straight a equals 2 space and space straight b equals 1.
Hence comma space correct space option space is space left parenthesis straight a right parenthesis.

Solution 7

begin mathsize 11px style table attributes columnalign left end attributes row cell cube root of 500 equals left parenthesis 500 right parenthesis to the power of bevelled 1 third end exponent equals open parentheses fraction numerator 500 space straight x space 2 over denominator 2 end fraction close parentheses to the power of bevelled 1 third end exponent space equals space open parentheses 1000 over 2 close parentheses to the power of bevelled 1 third end exponent equals space left parenthesis 10 to the power of up diagonal strike 3 end exponent right parenthesis to the power of bevelled fraction numerator 1 over denominator up diagonal strike 3 end fraction end exponent space. space 1 over 2 to the power of bevelled 1 third end exponent rightwards double arrow space 10 space. space 2 to the power of bevelled fraction numerator negative 1 over denominator 3 end fraction end exponent end cell row cell table attributes columnalign left end attributes row cell The space simplest space Rationalisation space factor space of space cube root of 500 straight space end cell row cell after space simplify space it space to space left parenthesis 10.2 to the power of negative bevelled 1 third end exponent right parenthesis space space is space space 2 to the power of bevelled 1 third end exponent space or space cube root of 2 space. end cell end table
Hence comma space correct space option space is space left parenthesis straight a right parenthesis. space space space space space space space end cell end table
end style


Solution 8

begin mathsize 11px style Rationalising straight space factor straight space of straight space any straight space number straight space of straight space kind straight space square root of straight a plus-or-minus square root of straight b space space space is space space square root of straight a minus-or-plus square root of straight b
So comma space for space given space number space square root of 3 plus square root of 5 comma space Rationalising space factor space would space be space square root of 3 minus square root of 5.
Hence comma space correct space option space is space left parenthesis straight c right parenthesis. end style

Solution 9

Error converting from MathML to accessible text.

Solution 10

begin mathsize 11px style table attributes columnalign left end attributes row cell straight x equals fraction numerator 2 over denominator 3 plus square root of 7 end fraction equals space fraction numerator 2 over denominator 3 plus square root of 7 end fraction cross times space fraction numerator 3 minus square root of 7 over denominator 3 minus square root of 7 end fraction equals fraction numerator 2 left parenthesis 3 minus square root of 7 right parenthesis over denominator left parenthesis 3 right parenthesis squared minus left parenthesis square root of 7 right parenthesis squared end fraction equals fraction numerator 6 minus 2 square root of 7 over denominator 9 minus 7 end fraction equals fraction numerator 6 minus 2 square root of 7 over denominator 2 end fraction equals 3 minus 2 square root of 7 end cell row blank row cell Now space space space left parenthesis straight x minus 3 right parenthesis squared equals left parenthesis up diagonal strike 3 minus square root of 7 minus up diagonal strike 3 right parenthesis squared equals open parentheses negative square root of 7 close parentheses squared equals 7 end cell end table
Hence comma space correct space option space is space left parenthesis straight d right parenthesis. end style

Solution 11

begin mathsize 11px style table attributes columnalign left end attributes row cell straight x equals 7 plus 4 square root of 3 comma space space space space xy equals 1 rightwards double arrow straight y equals 1 over straight x end cell row cell therefore space straight y equals straight space fraction numerator 1 over denominator 7 plus 4 square root of 3 end fraction end cell row cell therefore space straight y space equals space fraction numerator 1 over denominator 7 plus 4 square root of 3 end fraction cross times fraction numerator 7 minus 4 square root of 3 over denominator 7 minus 4 square root of 3 end fraction equals fraction numerator 7 minus 4 square root of 3 over denominator left parenthesis 7 right parenthesis squared minus left parenthesis 4 square root of 3 right parenthesis squared end fraction equals fraction numerator 7 minus 4 square root of 3 over denominator 49 minus 48 end fraction equals 7 minus 4 square root of 3 end cell row blank row cell Now comma space space 1 over straight x squared plus 1 over straight y squared equals fraction numerator straight y squared plus straight x squared over denominator straight x squared straight y squared end fraction equals fraction numerator straight x squared plus straight y squared over denominator left parenthesis xy right parenthesis squared end fraction end cell row cell straight x to the power of straight 2 equals left parenthesis 7 plus 4 square root of 3 right parenthesis squared equals 49 plus 48 plus 56 square root of 3 equals 97 plus 56 square root of 3 end cell row cell straight y to the power of straight 2 equals straight space left parenthesis 7 minus 4 square root of 3 right parenthesis squared equals space 49 plus 48 minus 56 square root of 3 equals 97 minus 56 square root of 3 end cell row cell therefore space straight x to the power of straight 2 plus straight y squared equals 97 plus 56 square root of 3 plus 97 minus 56 square root of 3 equals 194 end cell row cell xy equals 1 end cell end table

therefore space fraction numerator straight x squared plus straight y squared over denominator left parenthesis xy right parenthesis squared end fraction equals space 194 over left parenthesis 1 right parenthesis squared equals 194
Hence comma space correct space option space is space left parenthesis straight c right parenthesis. end style

Solution 12

begin mathsize 11px style table attributes columnalign left end attributes row cell straight x plus square root of 15 equals 4 end cell row cell rightwards double arrow straight x equals space 4 space minus space square root of 15 rightwards double arrow 1 over straight x equals fraction numerator 1 over denominator 4 minus square root of 15 end fraction end cell row cell 1 over straight x equals straight space fraction numerator 1 over denominator 4 minus square root of 15 end fraction cross times straight space fraction numerator 4 plus square root of 15 over denominator 4 plus square root of 15 end fraction equals fraction numerator 4 plus square root of 15 over denominator left parenthesis 4 right parenthesis squared minus left parenthesis square root of 15 right parenthesis squared end fraction equals fraction numerator 4 plus square root of 15 over denominator 16 minus 15 end fraction equals 4 plus square root of 15 end cell row blank row cell Now comma space straight x space plus space 1 over straight x equals 4 minus square root of 15 plus 4 plus square root of 15 equals 8 end cell row cell Hence comma space correct space option space is space left parenthesis straight c right parenthesis. end cell end table end style

Solution 13

Error converting from MathML to accessible text.

Solution 14

begin mathsize 11px style table attributes columnalign left end attributes row cell straight x space equals space fraction numerator square root of 3 minus square root of 2 over denominator square root of 3 plus square root of 2 end fraction end cell row cell therefore space straight x space equals space fraction numerator square root of 3 minus square root of 2 over denominator square root of 3 plus square root of 2 end fraction straight space straight x space fraction numerator square root of 3 minus square root of 2 over denominator square root of 3 minus square root of 2 end fraction space equals space fraction numerator left parenthesis square root of 3 minus square root of 2 right parenthesis squared over denominator 3 minus 2 end fraction equals space 5 space minus space 2 square root of 6 end cell row cell straight y space equals space fraction numerator square root of 3 plus square root of 2 over denominator square root of 3 minus square root of 2 end fraction space end cell row cell therefore space straight y equals space fraction numerator square root of 3 plus square root of 2 over denominator square root of 3 minus square root of 2 end fraction straight space straight x space fraction numerator square root of 3 plus square root of 2 over denominator square root of 3 plus square root of 2 end fraction space equals space fraction numerator left parenthesis square root of 3 plus square root of 2 right parenthesis squared over denominator 3 minus 2 end fraction equals space 5 space plus space 2 square root of 6 end cell end table

Now comma space straight x to the power of straight 2 space plus space xy space plus space straight y to the power of straight 2
equals left parenthesis 5 space minus space 2 square root of 6 right parenthesis squared plus left parenthesis 5 space minus space 2 square root of 6 right parenthesis left parenthesis 5 space plus space 2 square root of 6 right parenthesis plus left parenthesis 5 space plus space 2 square root of 6 right parenthesis squared
equals open parentheses 25 plus 24 minus 20 square root of 6 close parentheses plus open parentheses 25 minus 24 close parentheses plus open parentheses 25 plus 24 plus 20 square root of 6 close parentheses
equals 49 minus 20 square root of 6 plus 1 plus 49 plus 20 square root of 6
equals 9
Hence comma space correct space option space is space left parenthesis straight b right parenthesis.

end style

Solution 15

begin mathsize 11px style fraction numerator 1 over denominator square root of 9 minus square root of 8 end fraction
equals fraction numerator 1 over denominator square root of 9 minus square root of 8 end fraction straight space straight x straight space fraction numerator square root of 9 space plus space square root of 8 straight space over denominator square root of 9 space plus space square root of 8 end fraction
fraction numerator square root of 9  +  square root of 8 over denominator open parentheses square root of 9 close parentheses squared minus open parentheses square root of 8 close parentheses squared end fraction
equals fraction numerator square root of 9 plus square root of 8 over denominator 9 minus 8 end fraction
equals square root of 9 straight space plus straight space square root of 8
equals 3 plus 2 square root of 2
Hence comma space correct space option space is space left parenthesis straight a right parenthesis. end style

Solution 16

begin mathsize 11px style table attributes columnalign left end attributes row cell square root of 48 equals text   end text square root of 16 text  x 3 end text end root text  = 4 end text square root of 3 end cell row cell square root of 32 equals text   end text square root of 16 text  x 2 end text end root text  = 4 end text square root of 2 end cell row cell square root of 27 equals text   end text square root of text 9 x 3 end text end root text  = 3 end text square root of 3 end cell row cell square root of 18 equals text   end text square root of text 9 x 2 end text end root text  = 3 end text square root of 2 end cell row cell Now comma text    end text fraction numerator square root of 48 text  +  end text square root of 32 over denominator square root of 27 text  +  end text square root of 18 end fraction text  =  end text fraction numerator 4 square root of 3 text  + 4 end text square root of 2 over denominator 3 square root of 3 text  + 3 end text square root of 2 end fraction end cell row cell text        =  end text fraction numerator 4 text    end text left parenthesis square root of up diagonal strike 3 end root text + end text square root of up diagonal strike 2 end root right parenthesis over denominator 3 text    end text left parenthesis square root of up diagonal strike 3 end root text + end text square root of up diagonal strike 2 end root right parenthesis end fraction end cell row cell text        =   end text 4 over 3 end cell end table
Hence comma space correct space option space is space left parenthesis straight a right parenthesis. end style

RD Sharma Solution for Class 9 Mathematics Chapter 3 - Rationalisation Page/Excercise 3.18

Solution 17

begin mathsize 11px style fraction numerator 5 minus square root of 3 over denominator 2 plus square root of 3 end fraction
equals fraction numerator 5 minus square root of 3 over denominator 2 plus square root of 3 end fraction cross times fraction numerator 2 minus square root of 3 over denominator 2 minus square root of 3 end fraction
equals fraction numerator open parentheses 5 minus square root of 3 close parentheses open parentheses 2 minus square root of 3 close parentheses over denominator open parentheses 2 close parentheses squared minus open parentheses square root of 3 close parentheses squared end fraction
equals fraction numerator 10 minus 5 square root of 3 minus 2 square root of 3 plus 3 over denominator 4 minus 3 end fraction
equals fraction numerator 13 minus 7 square root of 3 over denominator 1 end fraction
equals 13 minus 7 square root of 3
rightwards double arrow straight x equals 13 space and space straight y equals negative 7
Hence comma space correct space option space is space left parenthesis straight a right parenthesis. end style

Solution 18

begin mathsize 11px style table attributes columnalign left end attributes row cell straight x equals straight space cube root of 2 plus square root of 3 end root  +  left parenthesis 2 plus square root of 3 right parenthesis to the power of bevelled 1 third end exponent end cell row cell straight x cubed straight space equals left curly bracket left parenthesis 2 plus square root of 3 right parenthesis to the power of bevelled fraction numerator 1 over denominator up diagonal strike 3 end fraction end exponent right curly bracket to the power of up diagonal strike 3 end exponent equals left parenthesis 2 plus square root of 3 right parenthesis end cell row cell rightwards double arrow 1 over straight x cubed equals fraction numerator 1 over denominator 2 plus square root of 3 end fraction equals fraction numerator 1 over denominator 2 plus square root of 3 end fraction cross times straight space fraction numerator 2 minus square root of 3 over denominator 2 minus square root of 3 end fraction equals fraction numerator 2 minus square root of 3 over denominator 4 minus 3 end fraction straight space equals 2 minus square root of 3 end cell row blank row cell Now comma space straight x to the power of straight 3 plus straight space 1 over straight x cubed equals straight space 2 plus square root of 3  + space 2 space minus space square root of 3 equals 4 end cell row cell space space space space space space space space space space space space space space space space space space space space space space end cell end table
Hence comma space correct space option space is space left parenthesis straight b right parenthesis. end style

Solution 19

begin mathsize 11px style square root of 3 minus 2 square root of 2 end root
equals square root of 2 plus 1 minus straight space 2 square root of 2 end root
equals square root of open parentheses square root of 2 close parentheses squared plus left parenthesis 1 right parenthesis squared minus 2 open parentheses square root of 2 close parentheses left parenthesis 1 right parenthesis end root
equals square root of open parentheses square root of 2 minus 1 close parentheses squared end root
equals square root of 2 minus 1
Hence comma space correct space option space is space left parenthesis straight a right parenthesis.
end style

Solution 20

begin mathsize 11px style square root of 5 plus 2 square root of 6 end root
equals square root of 3 plus 2 space plus space 2 open parentheses square root of 3 close parentheses open parentheses square root of 2 close parentheses end root
equals square root of open parentheses square root of 3 close parentheses squared plus open parentheses square root of 2 close parentheses squared plus 2 open parentheses square root of 3 close parentheses open parentheses square root of 2 close parentheses end root
equals square root of open parentheses square root of 3 plus square root of 2 close parentheses squared end root
equals square root of 3 plus square root of 2
Hence comma space correct space option space is space left parenthesis straight b right parenthesis. end style

Solution 21

begin mathsize 11px style text By Rationalising   end text fraction numerator square root of 2 minus 1 over denominator square root of 2 plus 1 end fraction text , end text space we space get
table attributes columnalign left end attributes row cell fraction numerator square root of 2 minus 1 over denominator square root of 2 plus 1 end fraction space straight x space fraction numerator square root of 2 minus 1 over denominator square root of 2 minus 1 end fraction  =  fraction numerator left parenthesis square root of 2 minus 1 right parenthesis squared over denominator left parenthesis square root of 2 right parenthesis squared minus 1 squared end fraction equals straight space left parenthesis square root of 2 minus 1 right parenthesis squared over 1 end cell row cell so straight space square root of fraction numerator square root of 2 minus 1 over denominator square root of 2 plus 1 end fraction end root space equals space square root of left parenthesis square root of 2 minus 1 right parenthesis squared over 1 end root  =  left parenthesis square root of 2 minus 1 right parenthesis equals 1.4142 minus 1 equals 0.4142 end cell end table
Hence comma space correct space option space is space left parenthesis straight c right parenthesis. end style

Solution 22

begin mathsize 11px style square root of 6 minus square root of 3
equals straight space square root of 3 left parenthesis square root of 2 minus 1 right parenthesis
table attributes columnalign left end attributes row cell Now comma space square root of 3 equals   1.732 space end cell row cell      space space space space space square root of 2 equals   1.414 end cell row cell therefore space square root of 6 minus square root of 3 space equals 1.732 left parenthesis 1.414 minus 1 right parenthesis equals 1.732   left parenthesis 0.414 right parenthesis equals space 0.717 space left parenthesis upto space 3 space decimal space places right parenthesis end cell end table

Hence comma space correct space option space is space left parenthesis straight b right parenthesis. end style

Solution 23

begin mathsize 11px style square root of 7 plus square root of 48 end root
equals square root of 7 plus 2 square root of 12 end root
equals square root of 4 space plus 3 space plus space 2 square root of 4 space straight x space square root of 3 end root
equals square root of open parentheses square root of 4 close parentheses squared plus open parentheses square root of 3 close parentheses squared plus space 2 space straight x space square root of 4 space straight x space square root of 3 end root
equals square root of open parentheses square root of 4 plus square root of 3 close parentheses squared end root
equals plus-or-minus straight space open parentheses square root of 4 plus square root of 3 close parentheses
Positve space value space is space square root of 4 plus straight space square root of 3 equals 2 plus square root of 3
Hence comma space correct space option space is space left parenthesis straight c right parenthesis. end style

Solution 24

begin mathsize 11px style straight x squared plus 1 over straight x squared minus 2 equals text   end text open parentheses straight x minus 1 over straight x close parentheses squared
straight x equals text   end text square root of 6 plus square root of 5
rightwards double arrow 1 over straight x text  =  end text fraction numerator 1 over denominator square root of 6 plus square root of 5 end fraction equals text   end text fraction numerator 1 over denominator square root of 6 plus square root of 5 end fraction text   end text straight x text   end text fraction numerator square root of 6 minus square root of 5 over denominator square root of 6 minus square root of 5 end fraction text = end text fraction numerator square root of 6 minus square root of 5 over denominator 1 end fraction equals square root of 6 minus square root of 5
Now comma
open parentheses straight x minus 1 over straight x close parentheses squared equals open square brackets square root of 6 plus square root of 5 minus open parentheses square root of 6 minus square root of 5 close parentheses close square brackets squared equals open parentheses 2 square root of 5 close parentheses squared equals 4 cross times 5 equals 20
Hence comma space correct space option space is space left parenthesis straight d right parenthesis. end style

Solution 25

begin mathsize 11px style square root of 13 minus straight a square root of 10 end root equals text   end text square root of 8 plus square root of 5
Squaring space both space sides comma space we space get
table attributes columnalign left end attributes row cell 13 minus straight a square root of 10 equals text   end text 8 plus 5 plus 2 square root of 40 end cell row cell rightwards double arrow 1 up diagonal strike 3 minus straight a square root of 10 text  -  end text 1 up diagonal strike 3 text  = 2 x 2 end text square root of 10 end cell end table
rightwards double arrow negative straight a square root of 10 equals 4 square root of 10
rightwards double arrow straight a equals negative 4
Hence comma space correct space option space is space left parenthesis straight c right parenthesis.
end style

TopperLearning provides step-by-step solutions for each question in each chapter in the RD Sharma textbook for class 9. Access the CBSE Class 9 Mathematics Chapter 3 - Rationalisation for free. The textbook questions have been solved by our subject matter experts to help you understand how to answer them. Our RD Sharma Textbook Solutions will help you to study and revise, and you can easily clear your fundamentals of Chapter 3 - Rationalisation.

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CBSE IX - Mathematics

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