Worksheet Solutions: Cubes and Cube Roots | Improve Your Calculations: Vedic Maths (English) - Class 6 PDF Download

Q1: 11³.

Ans: Here a = 1 , b = 1 ,  a³ = 1 ,  b/a = 1/1 = 1  (Here common ratio is equal to 1)
Now see the formation of the table:
First Row             1     1     1     1
Second Row               2     2
                    -------------------------
Add                     1     3     3     1
11³ = 1331

Q2: 13³.

Ans: Here a = 1 , b = 3 ,  a³ = 1 ,  b/a = 3/1 = 3 (Here common ratio is greater than 1)
Now see the formation of the table:
First Row              1     3     9     27  (Note: 4th term is just  b³  as shown above in algebraic  
Second Row                6     18               expression)
                          -------------------------
Add                      1     9     27    27
                          -------------------------
                         =   1     9      7     7       (Apply carry over rule)
                                   2     2
 =   1      9      9     7
                    2
                 =   1      1     9     7
                                    1
 =   2      1     9     7
13³ = 2197 

Q3: 52³.

Ans: Here a = 5 , b = 2 ,  a3 = 125 ,  b/a = 2/5 (Here common ratio is less than 1)
Now see the formation of the table:
First Row                 125     50     20     8
Second Row                     100     40
                           ---------------------------
Add                         125   150     60     8
                           ---------------------------
                           =  125       0        0     8        (Apply carry over rule)
                                        15        6
   =  140      6    0    8
52³ = 140608

Q4: 55³.

Ans: 1.  We consider 5 as 1st term and 5 as 2nd term, here both the digit is same so we take any one digit. Cube of 5 =125 and write 4 times
i.e. 125   125   125   125
2. In 2nd row double the 2 middle terms (. i.e. is 2nd term and 3rd term) & write just below 2nd & 3rd term.
                           125    125   125   125
                                      250    250                 (Doubled the value)
4.  Add them vertically in column. carry forward the 10th places digit to next column
                          125    125    125    125
                     +               250    250
                            41        38      12                (Carried forward)
                        166           3        7          5    (Answer)
             (55)³ = 166,375

Q5: 32³.

Ans: 1.  We consider 3 as 1st term and 2 as 2nd term, Cube the 1st term and 2nd term
i.e.       (3)³ = 27     and         (2)³ = 8
                               27                                  8
2.  Square the 1st   term i.e. 3² = 9 then multiply by 2nd term i.e. 9 × 2 = 18
                               27         18                   8
3. Square the 2nd term i.e. 2² = 4 then multiply by 1st term i.e. 4 × 3 = 12
                              27          18      12       8
4. In 2nd row double the 2 middle terms (. i.e. is 2nd term and 3rd term) & write just below 2nd & 3rd term.
                             27          18       12      8
                                            36       24               (Doubled the value)
5.  Add them vertically in column. carry forward the 10th places digit to next column
                             27         18       12      8
                    +                     36       24                
                               5           3                             (Carried forward)
                           32           7          6      8       (Answer)
             (32)³ = 32,768

Q6: Find Cube Root of 13824.

Ans: Step 1:
Identify the last three digits and make groups of three digits from right side. That is 13824 can be written as
13,   824
Step 2: 
Take the last group which is 824.  The last digit of 824 is 4.
Remember point 3, If the last digit of the perfect cube = 4, the last digit of the cube root = 4
Hence the right most digit of the cube root  = 4
Step 3:
Take the next group which is 13.
From point 3, we see that 13 lies between 8 and 27 which are cubes of 2 and 3 respectively. So we will take the cube root of the smaller number i.e. 8 which is 2.
So 2 is the tens digit of the answer.
We are done and the answer is '24'

Q7: Find Cube Root of 185193.

Ans: Step 1:
185193 can be written as
185,   193
Step 2:
Take the last group which is 193.  The last digit of 193 is 3.
Remember point 3, If the last digit of the perfect cube = 3, the last digit of the cube root = 7
Hence the right most digit of the cube root  = 7
Step 3:
Take the next group which is 185.
From point 3, we see that 185 lies between 125 and 216 which are cubes of 5 and 6 respectively. So we will take the cube root of the smaller number i.e. 125 which is 5.
So 5 is the tens digit of the answer.
So, the answer = 57.

Q8: Find the Cube Root of 32768 using Vedic Maths Method.

Ans: Step 1: Divide the number into two groups (Divide it in such a way, 2 digits in left and 3 digits in right.
32 | 768
Therefore,
L.H.S = 32 R.H.S = 768
Step 2: By observing the number 32 lies between cube root of 3 and 4 (i.e) 27 and 64 from table 1. we should take the cube root of the smaller number. so,we can conclude that cube root of the number should be 3.
Hence for first digit on L.H.S, the cube root is 3
Step 3: Coming to the R.H.S, from rule 2 it is shown as when the last digit of the cube is 8, its cube root should be 2.
Therefore, R.H.S of the cube root is 2
Finally, cube root for 32768 is 32.

Q9: Find the cube root of 287496.

Ans: We shall represent the number 287496 as 287 496
Next, we observe that the cube 287496 ends with a 6 and we know that when the cube ends with a 6, the cube root also ends with a 6. Thus our answer at this stage is 6. We have thus got the RHS of our answer.
To find the LHS of the answer we take the number which lies to the left of the slash. In this case, the number lying to the left of the slash is 287. Now, we need to find two perfect cubes between which the number 287 lies in the number line. From the key, we find that 287 lies between the perfect cubes 216 (the cube of 6) and 343 (the cube of 7).
Now, out of the numbers obtained above, we take the smaller number and put it on the LHS of the answer. Thus, out of 6 and 7, we take the smaller number 6 and put it beside the answer of 6 already obtained. Our final answer is 66. Thus, 66 is the cube root of 287496.

Q10: Find the cube root of 681472.

Ans: We represent 681472 as 681/472
The cube ends with a 2, so the root ends with an 8. The answer at this stage is 8.
681 lies between 512 (the cube of 8) and 729 (the cube of 9).
The smaller number is 8 and hence our final answer is 88.