cubes & cube roots
Chapter 17
CUBES & CUBE ROOTS Before learning Cubes and Cube Roots of numbers, one must be thorough with the following chart: Number 13 23 33 43 53 63 73 83 93
Cube 1 8 27 64 125 216 343 512 729
Last Digit 1 8 7 4 5 6 3 2 9
Memorizing the Cubes can be easy. But while memorizing the Last Digits of each cube one must note that the last digits of 13 =1, 43 =4, 53 =5, 63 =6, & 93 =9. (All these are marked in BOLD). The remaining four can be remembered in this manner: If the last digit is 8, (10 - 8 = 2), if the last digit is 7, (10 - 7 = 3). And now vice-versa. If the last digit is 2, (10 - 2 = 8) & if the last digit is 3, (10 - 3 = 7). Meaning Last Digits of 2, 3, 7 & 8 are difference of themselves from Perfect Base 10. Before starting with Cubes & Cube Roots, lets Study one more important aspect of Vedic Mathematics. Now some Algebra! One must recollect that in algebra, (a + b)3 = a3 + 3 a2 b + 3 a b2 + b3. Isn't it? That means the formula has 4 terms viz. a3 + 3 a2 b + 3 a b2 + b3. Similarly (a + b) 2 = a2 + 2 a b + b2. Right? What does (a + b)2 mean? It means (a + b) * (a + b). Now lets go back to General Multiplication studied earlier. (238)
How do you multiply a 2 digit number with another 2 digit number when not near a common base? VEDIC MATHS METHOD
ALGEBRAIC METHOD
Do you find any difference in both the methods other than alphabets in place of numbers? E To get the RHS part we multiply both the unit's place digits → 2*1 = 2. (In case of alphabets we multiply 'b' with 'b' to get b2 ). E To get the Middle part we multiply Crosswise and ADD the products → (3*1) + (2*2) = 7. (In case of alphabets too, we multiply Crosswise and ADD the products → (a * b) + (a * b) to get 2ab). E To get the LHS part we multiply both the ten's place digits → 3*2 = 6. (In case of alphabets we multiply 'a' with 'a' to get a2). E Finally to differentiate between the parts we use a slash and in algebra a '+' sign is used. After studying the above fact, one must have understood that Vedic Mathematics not only deals in Arithmetic, but it is equally Good in Algebraic Calculations! In fact there are many Amazing Vedic Mathematics methods that deals with algebra too. But again, covering Algebra in this book alone meant, altogether writing around 700 to 800 pages of accurate calculations. Therefore we have reserved Algebra for the future. (239)
Finding Cubes of Numbers e.g. 1: Find 243. As in algebra there are 4 terms ( a3 + 3 a2 b + 3 a b2 + b3 ) for (a + b)3, in Vedic Mathematics too we use 4 terms to find the Cube of a number.
Steps: 1. a = 2 and b = 4 of the given number. 2. The first term = a3 = Cube of 2 = 8. 3. Now we go on multiplying the present term with b / a, meaning 4/2 = 2/1 = 2. (In other words, we multiply the present term by the Ratio between ‘a’ & ‘b’). 4. To get 2nd Term = First Term multiplied by b/a or 8*2=16. 5. To get 3rd Term = Second Term multiplied by b/a or 16*2 = 32. 6. To get 4th and Final Term = Third Term multiplied by b/a or 32 * 2 = 64. 7. Note that the last Term should be equal to b3 = Cube of 4 = 64. Only then the in-between Terms are correct. 8. Now multiply the Middle two Terms by 3 (the Term in the Ten's place and the term in the Hundred's place) and write the derived answer. 9. Now ADD the terms in respect to the place value. And now to get the final answer, starting from the Right to Left, take one digit from each part as the answer and rest of the digits are carried over and added to the next left part, similar to what we do in General Multiplication. (240)
e.g. 2: Find 633.
Steps: 1. a = 6 and b = 3 of the given number. 2. The first term = a3 = Cube of 6 = 216. 3. Now we multiply the present term with b/a, meaning 3/6 = ½. 4. 2nd Term = First Term multiplied by b/a or 216 * ½ = 108. 5. 3rd Term = Second Term multiplied by b/a or 108*½ =54. 6. 4th Term = Third Term multiplied by b/a or 54 * ½ = 27 or simply b3 = Cube of 3 = 27. 7. Now multiply the Middle two Terms by 3 (the Term in the Ten's place and the term in the Hundred's place) and write the derived answer. 8. Now ADD the terms in respect to the place value as shown in the diagram above. And now to get the final answer, take one digit from each part as the answer and carry over and add the rest of the digits to the next left part, similar to what we do in General Multiplication. e.g. 3: Find 823.
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Steps: 1. a = 8 and b = 2 of the given number. 2. The first term = a3 = Cube of 8 = 512. Now, Present term multiplied by b / a, meaning 2/8 = ¼ . 3. 2nd Term = 512 * ¼ = 128. 4. 3rd Term = 128 * ¼ = 32. 5. 4th Term = 32 * ¼ = 8. (This Term should be equal to b3 = Cube of 2 = 8). 6. Now multiply the Middle two Terms by 3 (the Term in the Ten's place and the term in the Hundred's place) and write the derived answer. 7. ADD the terms in respect to the place value as shown in the diagram above. 8. Get the Final answer from the derived parts, as explained earlier. Cubing is not as easy as it seems. The last three examples intentionally had easy numbers. Meaning the value of b/a was very easy. When there isn't a clear ratio of a & b, the actual task of cubing begins. Lets study such examples now. e.g. 4: Find 533.
(242)
Steps: 1. a = 5 and b = 3 of the given number. 2. The first term = a3 = Cube of 5 = 125. Now, Present term multiplied by b / a, meaning 3/5. In this case ‘ / ’ should be read as “Divided By” / “Upon” sign. 3. 2nd Term = 125 * 3/5 = 75. 4. 3rd Term = 75 * 3/5 = 45. 5. 4th Term = 45 * 3/5 = 27. (This Term should be equal to b3 = Cube of 3 = 27). 6. Now multiply the Middle two Terms by 3 (the Term in the Ten's place and the term in the Hundred's place) and write the derived answer. 7. ADD the terms in respect to the place value as shown in the diagram above. 8. Get the Final answer from the derived parts, as explained earlier. e.g. 5: Find 673.
Steps: 1. a = 6 and b = 7 of the given number. 2. The first term = a3 = Cube of 6 = 216. Now, Present term multiplied by b / a, meaning 7/6. 3. 2nd Term = 216 * 7/6 = 252. 4. 3rd Term = 252 * 7/6 = 294. 5. 4th Term = 294 * 7/6 = 343. (This Term should be equal to b3 = Cube of 7 = 343). (243)
6. Now multiply the Middle two Terms by 3 (the Term in the Ten's place and the term in the Hundred's place) and write the derived answer. 7. ADD the terms in respect to the place value as shown in the diagram above. 8. Get the Final answer from the derived parts, as explained earlier. e.g. 6: Find 943.
Steps: 1. a = 9 and b = 4 of the given number. 2. The first term = a3 = Cube of 9 = 729. Now, Present term multiplied by b / a, meaning 4/9. 3. 2nd Term = 729 * 4/9 = 324. 4. 3rd Term = 324 * 4/9 = 144. 5. 4th Term = 144 * 4/9 = 64. (This Term should be equal to b3 = Cube of 4 = 64). 6. Now multiply the Middle two Terms by 3 (the Term in the Ten's place and the term in the Hundred's place) and write the derived answer. 7. ADD the terms in respect to the place value as shown in the diagram above. 8. Get the Final answer from the derived parts, as explained earlier.
(244)
e.g. 7: Find 373.
Steps: 1. a = 3 and b = 7 of the given number. 2. The first term = a3 = Cube of 3 = 27. Now, Present term multiplied by b / a, meaning 7/3. 3. 2nd Term = 27 * 7/3 = 63. 4. 3rd Term = 63 * 7/3 = 147. 5. 4th Term = 147 * 7/3 = 343. This Term should be equal to b3. 6. Now multiply the Middle two Terms by 3 (the Term in the Ten's place and the term in the Hundred's place) and write the derived answer. 7. ADD the terms in respect to the place value as shown in the diagram above. 8. Get the Final answer from the derived parts, as explained earlier. e.g. 8: Find 443.
Steps: 1. a = 4 and b = 4 of the given number. 2. The first term = a3 = Cube of 4 = 64. Now, b / a = 4/4 = 1 or same terms. (245)
3. 4. 5. 6.
2nd Term = 64 * 1 = 64. 3rd Term = 64 * 1 = 64. 4th Term = 64 * 1 = 64. This Term should be equal to b3. The Rest is as usual.
Finding Cube Roots of Numbers Before proceeding one should be thorough with the earlier chart given on page No.238. If the chart is memorized, meaning if one identifies the corresponding number to a last digit fast, one can find the Cubes Roots Fast. How Fast? Perhaps 3 to 5 seconds! Yes, seconds and not minutes. e.g. 1: Find Cube Root (CR) of 175616. CR = 56. Steps: 1. Split the given number into three digits each starting from Unit's place = 175616. Remember that the L.H.S. Part can have 1, 2 or 3 digits. 2. 175 / 616. Splitting is not necessarily on paper. It can be done mentally too! 3. RHS = Find the Last Digit of the group 616. Its 6. And 6 is the Last Digit of 63. Hence the Cube Roots' Last Digit is 6. 4. LHS = 175. From this number, the Greatest Cube of what number can be subtracted? Meaning from 175 can we subtract 63, which is 216? NO. From 175 we can only subtract cube of 5, which is 125. Because we can subtract only the cube of 5, that's the Greatest Cube which can be subtracted from 175. Hence Cube Roots' LHS digit is 5. 5. Therefore CR of 175616 = 56. (246)
Lets study some more examples to have a better understanding of this method. e.g. 2: Find Cube Root (CR) of 592704. CR = 84. Steps: 1. Split 592704 into 2 groups of three digits each (592 / 704), starting from the Unit's place. 2. The Left Group will either have 1, 2 or 3 digits. 3. RHS = Last Digit of the RHS group 704 = 4. And 4 is the Last Digit of 43. Hence the Cube Roots' RHS Digit = 4. 4. LHS = 592. From this number, can we subtract 93, which is 729? NO. From 592 the Last Cube that can be subtracted is 83, which is 512. Hence Cube Roots' LHS digit is 8. 5. Therefore CR of 592704 = 84. e.g. 3: Find Cube Root (CR) of 42875. CR = 35. Steps: 1. Split 42875 into 2 groups of three digits each (42 / 875), starting from the Unit's place. 2. Note that the Left Group has only 2 digits. 3. RHS = Last Digit of the RHS group 875 = 5. And 5 is the Last Digit of 53. Hence the Cube Roots' RHS Digit = 5. 4. LHS = 42. Cube of what number can be subtracted from 42? Can we subtract 53? NO. Can we subtract 43? NO. Can we subtract 33? YES! Hence Cube Roots' LHS digit is 3. 5. Therefore CR of 42875 = 35. (247)
e.g. 4: Find Cube Root (CR) of 6859. CR = 19. Steps: 1. Split 6859 into 2 groups of three digits each (6 / 859), starting from the Unit's place. 2. Note that the Left Group has only 1 digit. 3. RHS = Last Digit of the RHS group 859 = 9. And 9 is the Last Digit of 93. Hence the Cube Roots' RHS Digit = 9. 4. LHS = 6. Cube of what number can be subtracted from 6? Can we subtract 33? No. Can we subtract 23? NO. Can we subtract 13? YES! Hence Cube Roots' LHS digit is 1. 5. Therefore CR of 6859 = 19. e.g. 5: Find Cube Root (CR) of 19683. CR = 27. Steps: 1. Split 19683 into 2 groups of three digits each (19 / 683), starting from the Unit's place. 2. Note that the Left Group has only 2 digit. 3. RHS = Last Digit of the RHS group 683 = 3. And 3 is the Last Digit of 73. Hence the Cube Roots' RHS Digit = 7. Note the change here. If the Last Digit of the given number is 3, the CR's Last digit will be 7. If the Last digit of the given number is either 2, 8, 3 or 7, we have to think in terms of 10 minus the number. This is explained earlier. 4. LHS = 19. Cube of what number can be subtracted from 19? Can we subtract 33? No. Can we subtract 23? YES! Hence Cube Roots' LHS digit is 2. 5. Therefore CR of 19683 = 27. (248)
e.g. 6: Find Cube Root (CR) of 250047. CR = 63. Steps: 1. Split 250047 into 2 groups of three digits each (250 / 047), starting from the Unit's place. 2. RHS = Last Digit of the RHS group 047 = 7. And 7 is the Last Digit of 33. Hence the Cube Roots' RHS Digit = 3. 3. LHS = 250. Cube of what number can be subtracted from 250? Can we subtract 73? No. Can we subtract 63? YES! Hence Cube Roots' LHS digit is 6. 4. Therefore CR of 250047 = 63. e.g. 7: Find Cube Root (CR) of 110592. CR = 48. Steps: 1. Split 110592 into 2 groups of three digits each (110 / 592), starting from the Unit's place. 2. RHS = Last Digit of the RHS group 592 = 2. And 2 is the Last Digit of 83. Hence the Cube Roots' RHS Digit = 8. 3. LHS = 110. Cube of what number can be subtracted from 110? Can we subtract 63? No. Can we subtract 53? NO. Can we subtract 43? YES! Hence CR's LHS digit is 4. 4. Therefore CR of 110592 = 48. Now, is there anyone around? We literally mean it. Is anyone around? May be a friend or a relative. Hand over a calculator to that person and ask him / her to choose any two digit number and Cube it. Of course he / she does not reveal the chosen 2 digit number to you. You are only told the cube. You have to find the Cube Root by the just studied method. Calculate how long (249)
you take to identify the Cube Root of the number given by him / her. Improve your time with every try. Memorizing the chart on page 238 will be of great help. Initially, one should be taking not more than 10 seconds. e.g. 8: Assume the Cube told to you by your friend / relative is 157464. Steps to do it mentally: 1. Last digit of the given number is 4 hence Cube Root's last digit = 4. 2. The given number has 6 digits, so obviously 157 minus Greatest Cube of what number on the LHS. 3. Can you subtract 63? NO. Can you subtract 53? YES! Hence CR's LHS digit is 5. 4. Therefore CR of 157464 = 54. That's it. More and more practice will improve your speed. e.g. 9: Find Cube Root of 456533. Steps to do it mentally: 1. Last digit of the given number is 3 hence Cube Root's last digit = 7. 2. The given number has 6 digits, so obviously 456 minus Greatest Cube of what number on the LHS. 3. Can we subtract 83? NO. Can you subtract 73? YES! Hence CR's LHS digit is 7. 4. Therefore CR of 456533 = 77.
(250)
e.g. 10: Find Cube Root of 68921. Steps to do it mentally: 1. Last digit of the given number is 1 hence Cube Root's last digit = 1. 2. The given number has 5 digits, so obviously 68 minus Greatest Cube of what number on the LHS. 3. Can we subtract 53? NO. Can you subtract 43? YES! Hence CR's LHS digit is 4. 4. Therefore CR of 68921 = 41. Now do you wonder why this method is not taught in schools? Why the long LCM / Factorization method? We too can't find an answer. Its just a matter of time. We wish sooner or later all schools across the world should go the Vedic Mathematics way! Now lets learn how to find the Cube Roots of Cubes having 7 or more digits. Meaning the Cube Roots will be of 3 digits.
e.g. 1: Find Cube Root (CR) of 1 8 6 0 8 6 7. Now mind you, you can't get this in 3, 5 or 10 seconds. You require slightly more time. Steps: 1. Split 1860867 into groups of three digits each (1 / 860 / 867), starting from the Unit's place. 2. Since 3 groups are formed, it indicates that the CR is of 3 digits. Yes! No. of groups = No. of digits in the CR. 3. Hence the answer will be in terms of 'a', 'b' and 'c'. The RHS digit = c, the LHS digit = a, and the middle digit = b. 4. 'a' and 'c' digits are found by the same method studied earlier. (251)
CUBES & CUBE ROOTS Before learning Cubes and Cube Roots of numbers, one must be thorough with the following chart: Number 13 23 33 43 53 63 73 83 93
Cube 1 8 27 64 125 216 343 512 729
Last Digit 1 8 7 4 5 6 3 2 9
Memorizing the Cubes can be easy. But while memorizing the Last Digits of each cube one must note that the last digits of 13 =1, 43 =4, 53 =5, 63 =6, & 93 =9. (All these are marked in BOLD). The remaining four can be remembered in this manner: If the last digit is 8, (10 - 8 = 2), if the last digit is 7, (10 - 7 = 3). And now vice-versa. If the last digit is 2, (10 - 2 = 8) & if the last digit is 3, (10 - 3 = 7). Meaning Last Digits of 2, 3, 7 & 8 are difference of themselves from Perfect Base 10. Before starting with Cubes & Cube Roots, lets Study one more important aspect of Vedic Mathematics. Now some Algebra! One must recollect that in algebra, (a + b)3 = a3 + 3 a2 b + 3 a b2 + b3. Isn't it? That means the formula has 4 terms viz. a3 + 3 a2 b + 3 a b2 + b3. Similarly (a + b) 2 = a2 + 2 a b + b2. Right? What does (a + b)2 mean? It means (a + b) * (a + b). Now lets go back to General Multiplication studied earlier. (238)
How do you multiply a 2 digit number with another 2 digit number when not near a common base? VEDIC MATHS METHOD
ALGEBRAIC METHOD
Do you find any difference in both the methods other than alphabets in place of numbers? E To get the RHS part we multiply both the unit's place digits → 2*1 = 2. (In case of alphabets we multiply 'b' with 'b' to get b2 ). E To get the Middle part we multiply Crosswise and ADD the products → (3*1) + (2*2) = 7. (In case of alphabets too, we multiply Crosswise and ADD the products → (a * b) + (a * b) to get 2ab). E To get the LHS part we multiply both the ten's place digits → 3*2 = 6. (In case of alphabets we multiply 'a' with 'a' to get a2). E Finally to differentiate between the parts we use a slash and in algebra a '+' sign is used. After studying the above fact, one must have understood that Vedic Mathematics not only deals in Arithmetic, but it is equally Good in Algebraic Calculations! In fact there are many Amazing Vedic Mathematics methods that deals with algebra too. But again, covering Algebra in this book alone meant, altogether writing around 700 to 800 pages of accurate calculations. Therefore we have reserved Algebra for the future. (239)
Finding Cubes of Numbers e.g. 1: Find 243. As in algebra there are 4 terms ( a3 + 3 a2 b + 3 a b2 + b3 ) for (a + b)3, in Vedic Mathematics too we use 4 terms to find the Cube of a number.
Steps: 1. a = 2 and b = 4 of the given number. 2. The first term = a3 = Cube of 2 = 8. 3. Now we go on multiplying the present term with b / a, meaning 4/2 = 2/1 = 2. (In other words, we multiply the present term by the Ratio between ‘a’ & ‘b’). 4. To get 2nd Term = First Term multiplied by b/a or 8*2=16. 5. To get 3rd Term = Second Term multiplied by b/a or 16*2 = 32. 6. To get 4th and Final Term = Third Term multiplied by b/a or 32 * 2 = 64. 7. Note that the last Term should be equal to b3 = Cube of 4 = 64. Only then the in-between Terms are correct. 8. Now multiply the Middle two Terms by 3 (the Term in the Ten's place and the term in the Hundred's place) and write the derived answer. 9. Now ADD the terms in respect to the place value. And now to get the final answer, starting from the Right to Left, take one digit from each part as the answer and rest of the digits are carried over and added to the next left part, similar to what we do in General Multiplication. (240)
e.g. 2: Find 633.
Steps: 1. a = 6 and b = 3 of the given number. 2. The first term = a3 = Cube of 6 = 216. 3. Now we multiply the present term with b/a, meaning 3/6 = ½. 4. 2nd Term = First Term multiplied by b/a or 216 * ½ = 108. 5. 3rd Term = Second Term multiplied by b/a or 108*½ =54. 6. 4th Term = Third Term multiplied by b/a or 54 * ½ = 27 or simply b3 = Cube of 3 = 27. 7. Now multiply the Middle two Terms by 3 (the Term in the Ten's place and the term in the Hundred's place) and write the derived answer. 8. Now ADD the terms in respect to the place value as shown in the diagram above. And now to get the final answer, take one digit from each part as the answer and carry over and add the rest of the digits to the next left part, similar to what we do in General Multiplication. e.g. 3: Find 823.
(241)
Steps: 1. a = 8 and b = 2 of the given number. 2. The first term = a3 = Cube of 8 = 512. Now, Present term multiplied by b / a, meaning 2/8 = ¼ . 3. 2nd Term = 512 * ¼ = 128. 4. 3rd Term = 128 * ¼ = 32. 5. 4th Term = 32 * ¼ = 8. (This Term should be equal to b3 = Cube of 2 = 8). 6. Now multiply the Middle two Terms by 3 (the Term in the Ten's place and the term in the Hundred's place) and write the derived answer. 7. ADD the terms in respect to the place value as shown in the diagram above. 8. Get the Final answer from the derived parts, as explained earlier. Cubing is not as easy as it seems. The last three examples intentionally had easy numbers. Meaning the value of b/a was very easy. When there isn't a clear ratio of a & b, the actual task of cubing begins. Lets study such examples now. e.g. 4: Find 533.
(242)
Steps: 1. a = 5 and b = 3 of the given number. 2. The first term = a3 = Cube of 5 = 125. Now, Present term multiplied by b / a, meaning 3/5. In this case ‘ / ’ should be read as “Divided By” / “Upon” sign. 3. 2nd Term = 125 * 3/5 = 75. 4. 3rd Term = 75 * 3/5 = 45. 5. 4th Term = 45 * 3/5 = 27. (This Term should be equal to b3 = Cube of 3 = 27). 6. Now multiply the Middle two Terms by 3 (the Term in the Ten's place and the term in the Hundred's place) and write the derived answer. 7. ADD the terms in respect to the place value as shown in the diagram above. 8. Get the Final answer from the derived parts, as explained earlier. e.g. 5: Find 673.
Steps: 1. a = 6 and b = 7 of the given number. 2. The first term = a3 = Cube of 6 = 216. Now, Present term multiplied by b / a, meaning 7/6. 3. 2nd Term = 216 * 7/6 = 252. 4. 3rd Term = 252 * 7/6 = 294. 5. 4th Term = 294 * 7/6 = 343. (This Term should be equal to b3 = Cube of 7 = 343). (243)
6. Now multiply the Middle two Terms by 3 (the Term in the Ten's place and the term in the Hundred's place) and write the derived answer. 7. ADD the terms in respect to the place value as shown in the diagram above. 8. Get the Final answer from the derived parts, as explained earlier. e.g. 6: Find 943.
Steps: 1. a = 9 and b = 4 of the given number. 2. The first term = a3 = Cube of 9 = 729. Now, Present term multiplied by b / a, meaning 4/9. 3. 2nd Term = 729 * 4/9 = 324. 4. 3rd Term = 324 * 4/9 = 144. 5. 4th Term = 144 * 4/9 = 64. (This Term should be equal to b3 = Cube of 4 = 64). 6. Now multiply the Middle two Terms by 3 (the Term in the Ten's place and the term in the Hundred's place) and write the derived answer. 7. ADD the terms in respect to the place value as shown in the diagram above. 8. Get the Final answer from the derived parts, as explained earlier.
(244)
e.g. 7: Find 373.
Steps: 1. a = 3 and b = 7 of the given number. 2. The first term = a3 = Cube of 3 = 27. Now, Present term multiplied by b / a, meaning 7/3. 3. 2nd Term = 27 * 7/3 = 63. 4. 3rd Term = 63 * 7/3 = 147. 5. 4th Term = 147 * 7/3 = 343. This Term should be equal to b3. 6. Now multiply the Middle two Terms by 3 (the Term in the Ten's place and the term in the Hundred's place) and write the derived answer. 7. ADD the terms in respect to the place value as shown in the diagram above. 8. Get the Final answer from the derived parts, as explained earlier. e.g. 8: Find 443.
Steps: 1. a = 4 and b = 4 of the given number. 2. The first term = a3 = Cube of 4 = 64. Now, b / a = 4/4 = 1 or same terms. (245)
3. 4. 5. 6.
2nd Term = 64 * 1 = 64. 3rd Term = 64 * 1 = 64. 4th Term = 64 * 1 = 64. This Term should be equal to b3. The Rest is as usual.
Finding Cube Roots of Numbers Before proceeding one should be thorough with the earlier chart given on page No.238. If the chart is memorized, meaning if one identifies the corresponding number to a last digit fast, one can find the Cubes Roots Fast. How Fast? Perhaps 3 to 5 seconds! Yes, seconds and not minutes. e.g. 1: Find Cube Root (CR) of 175616. CR = 56. Steps: 1. Split the given number into three digits each starting from Unit's place = 175616. Remember that the L.H.S. Part can have 1, 2 or 3 digits. 2. 175 / 616. Splitting is not necessarily on paper. It can be done mentally too! 3. RHS = Find the Last Digit of the group 616. Its 6. And 6 is the Last Digit of 63. Hence the Cube Roots' Last Digit is 6. 4. LHS = 175. From this number, the Greatest Cube of what number can be subtracted? Meaning from 175 can we subtract 63, which is 216? NO. From 175 we can only subtract cube of 5, which is 125. Because we can subtract only the cube of 5, that's the Greatest Cube which can be subtracted from 175. Hence Cube Roots' LHS digit is 5. 5. Therefore CR of 175616 = 56. (246)
Lets study some more examples to have a better understanding of this method. e.g. 2: Find Cube Root (CR) of 592704. CR = 84. Steps: 1. Split 592704 into 2 groups of three digits each (592 / 704), starting from the Unit's place. 2. The Left Group will either have 1, 2 or 3 digits. 3. RHS = Last Digit of the RHS group 704 = 4. And 4 is the Last Digit of 43. Hence the Cube Roots' RHS Digit = 4. 4. LHS = 592. From this number, can we subtract 93, which is 729? NO. From 592 the Last Cube that can be subtracted is 83, which is 512. Hence Cube Roots' LHS digit is 8. 5. Therefore CR of 592704 = 84. e.g. 3: Find Cube Root (CR) of 42875. CR = 35. Steps: 1. Split 42875 into 2 groups of three digits each (42 / 875), starting from the Unit's place. 2. Note that the Left Group has only 2 digits. 3. RHS = Last Digit of the RHS group 875 = 5. And 5 is the Last Digit of 53. Hence the Cube Roots' RHS Digit = 5. 4. LHS = 42. Cube of what number can be subtracted from 42? Can we subtract 53? NO. Can we subtract 43? NO. Can we subtract 33? YES! Hence Cube Roots' LHS digit is 3. 5. Therefore CR of 42875 = 35. (247)
e.g. 4: Find Cube Root (CR) of 6859. CR = 19. Steps: 1. Split 6859 into 2 groups of three digits each (6 / 859), starting from the Unit's place. 2. Note that the Left Group has only 1 digit. 3. RHS = Last Digit of the RHS group 859 = 9. And 9 is the Last Digit of 93. Hence the Cube Roots' RHS Digit = 9. 4. LHS = 6. Cube of what number can be subtracted from 6? Can we subtract 33? No. Can we subtract 23? NO. Can we subtract 13? YES! Hence Cube Roots' LHS digit is 1. 5. Therefore CR of 6859 = 19. e.g. 5: Find Cube Root (CR) of 19683. CR = 27. Steps: 1. Split 19683 into 2 groups of three digits each (19 / 683), starting from the Unit's place. 2. Note that the Left Group has only 2 digit. 3. RHS = Last Digit of the RHS group 683 = 3. And 3 is the Last Digit of 73. Hence the Cube Roots' RHS Digit = 7. Note the change here. If the Last Digit of the given number is 3, the CR's Last digit will be 7. If the Last digit of the given number is either 2, 8, 3 or 7, we have to think in terms of 10 minus the number. This is explained earlier. 4. LHS = 19. Cube of what number can be subtracted from 19? Can we subtract 33? No. Can we subtract 23? YES! Hence Cube Roots' LHS digit is 2. 5. Therefore CR of 19683 = 27. (248)
e.g. 6: Find Cube Root (CR) of 250047. CR = 63. Steps: 1. Split 250047 into 2 groups of three digits each (250 / 047), starting from the Unit's place. 2. RHS = Last Digit of the RHS group 047 = 7. And 7 is the Last Digit of 33. Hence the Cube Roots' RHS Digit = 3. 3. LHS = 250. Cube of what number can be subtracted from 250? Can we subtract 73? No. Can we subtract 63? YES! Hence Cube Roots' LHS digit is 6. 4. Therefore CR of 250047 = 63. e.g. 7: Find Cube Root (CR) of 110592. CR = 48. Steps: 1. Split 110592 into 2 groups of three digits each (110 / 592), starting from the Unit's place. 2. RHS = Last Digit of the RHS group 592 = 2. And 2 is the Last Digit of 83. Hence the Cube Roots' RHS Digit = 8. 3. LHS = 110. Cube of what number can be subtracted from 110? Can we subtract 63? No. Can we subtract 53? NO. Can we subtract 43? YES! Hence CR's LHS digit is 4. 4. Therefore CR of 110592 = 48. Now, is there anyone around? We literally mean it. Is anyone around? May be a friend or a relative. Hand over a calculator to that person and ask him / her to choose any two digit number and Cube it. Of course he / she does not reveal the chosen 2 digit number to you. You are only told the cube. You have to find the Cube Root by the just studied method. Calculate how long (249)
you take to identify the Cube Root of the number given by him / her. Improve your time with every try. Memorizing the chart on page 238 will be of great help. Initially, one should be taking not more than 10 seconds. e.g. 8: Assume the Cube told to you by your friend / relative is 157464. Steps to do it mentally: 1. Last digit of the given number is 4 hence Cube Root's last digit = 4. 2. The given number has 6 digits, so obviously 157 minus Greatest Cube of what number on the LHS. 3. Can you subtract 63? NO. Can you subtract 53? YES! Hence CR's LHS digit is 5. 4. Therefore CR of 157464 = 54. That's it. More and more practice will improve your speed. e.g. 9: Find Cube Root of 456533. Steps to do it mentally: 1. Last digit of the given number is 3 hence Cube Root's last digit = 7. 2. The given number has 6 digits, so obviously 456 minus Greatest Cube of what number on the LHS. 3. Can we subtract 83? NO. Can you subtract 73? YES! Hence CR's LHS digit is 7. 4. Therefore CR of 456533 = 77.
(250)
e.g. 10: Find Cube Root of 68921. Steps to do it mentally: 1. Last digit of the given number is 1 hence Cube Root's last digit = 1. 2. The given number has 5 digits, so obviously 68 minus Greatest Cube of what number on the LHS. 3. Can we subtract 53? NO. Can you subtract 43? YES! Hence CR's LHS digit is 4. 4. Therefore CR of 68921 = 41. Now do you wonder why this method is not taught in schools? Why the long LCM / Factorization method? We too can't find an answer. Its just a matter of time. We wish sooner or later all schools across the world should go the Vedic Mathematics way! Now lets learn how to find the Cube Roots of Cubes having 7 or more digits. Meaning the Cube Roots will be of 3 digits.
e.g. 1: Find Cube Root (CR) of 1 8 6 0 8 6 7. Now mind you, you can't get this in 3, 5 or 10 seconds. You require slightly more time. Steps: 1. Split 1860867 into groups of three digits each (1 / 860 / 867), starting from the Unit's place. 2. Since 3 groups are formed, it indicates that the CR is of 3 digits. Yes! No. of groups = No. of digits in the CR. 3. Hence the answer will be in terms of 'a', 'b' and 'c'. The RHS digit = c, the LHS digit = a, and the middle digit = b. 4. 'a' and 'c' digits are found by the same method studied earlier. (251)
