Cube Root by Successive Subtraction


 
 
Concept Explanation
 

Cube Root by Successive Subtraction

Like squares of natural numbers, cubes too have some interesting patterns.

1^{3}=1

2^{3}=8Rightarrow 2^{3}-1^{3}=7=1+1times6

                                       =1+2times1times3

3^{3}=27Rightarrow 3^{3}-2^{3}=27-8=19

                                      =1+1times6+2times6

                                      =1+3times2times3

4^{3}=64Rightarrow 4^{3}-3^{3}=64-27=37

                                        =1+1times6+2times6+3times6

                                         =1+4times3times3

5^{3}=125Rightarrow 5^{3}-4^{3}=125-64=61

                                           =1+1times6+2times6+3times6+4times6

                                            =1+5times4times3

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9^{3}=729Rightarrow 9^{3}-8^{3}=729-512=217

                                            =1+1times6+2times6+3times6+4times6...+8times6

                                          =1+9times8times3

Also                          1=1^{3}

                        1+7=2^{3}

              1+7+19=3^{3}

    1+7+19+37=4^{3}

1+7+19+...+217=9^{3}

From the above pattern, we see that 2^{3} is the sum of the first two numbers of the sequence 1,7,19.37,...Similarly, 3^{3} is the sum of the first three numbers and so on.In short,these numbers(1,7,19,...)may be obtained by putting n=1,2,3,...in 1+n(n-1) X 3

Thus to find the cube root of a given number, we go on subtracting the numbers of the sequence 1,7,19,37,... till we get a zero. The number of subtractions needed for this purpose is the cube root of the given number.

Illustration: Find the cube root of 343 by successive subtraction.

Solution: Subtract the numbers of the sequence 1,7,19,37,...from 343 till we get zero.

343-1=342

342-7=335

335-19=316

316-37=279

279-61=218

218-91=127

127-127=0

Since we have subtracted seven times to get 0, Hence sqrt[3]{343}=7     

Illustration: Find the cube root of 216 by successive subtraction.

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Sample Questions
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Question : 1

What is the number obtained in the eighth step while finding the cube root of 729 by successive subtraction?

Right Option : D
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Explanation
Question : 2

Find the cube root of 64 by successive subtraction of numbers 1,7,19,37,...

Right Option : A
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Explanation
Question : 3

What is the number obtained in the fourth step while finding the cube root of 512 by successive subtraction?

Right Option : C
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Explanation
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