Cubing of numbers near powers of 10
Explaining the method with the help of an example .
Example 1 : Find the cube of 104 .
Answer : -
Step 1 : -
In the above excess is 4 .
Find cube of 4 and retain two right most digit for answer . Remaining digits are for carry .
i.e.
43 = 64
Obtained digit = 64 Carry = 0
Step 2 : -
Double the previous excess . Find the new excess and multiply it with the previous excess .
i.e.
2 * 4 = 8
8 + 104 = 112 (Excess is 12)
Multiply this with the previous excess 4
12 * 4 = 48
Retain two right most digit for answer . Remaining digits are for carry .
Obtained digit = 48 64 Carry = 0
Step 3 : -
Now place the digit plus the double of excess to the left .
i.e.
112 4864
One Line representation
Way of writing = 112 / 048 / 064
Hence the cube of 104 is
1124864
Example 2 : - Find the cube of 106 .
Answer : -
Step 1 : -
Find the cube of excess .
63 = 216
Obtained digit = 16 Carry = 2
Step 2 : -
Double the excess . Add it to the original dgit . And multiply the new excess with the previous excess .
6 * 2 = 12 + 106 = 118 (New Excess = 18)
18 * 6 = 108
Add previous carry = 108 + 2 = 110
Obtained digit = 1016 Carry = 1
Step 3 : -
Add new excess(obtained in previous step) to the base , Add carry and then place it to the left of above obtained digit .
118 + 1 = 119
Obtained digit = 1191016
Hence the answer is 1191016 .
One Line representation of above question .
118 / 108 / 216
= 1191016
Example 3 : - Find the cubve of 94 .
Answer : -
Step 1 : -
Excess is - 6 .
Cube of -6 = - 216
Way of writing = / - 216
Step 2 : -
Double the excess . Add it to the original digit .
- 6 * 2 = - 12
94 - 12 = 82 (New excess is - 18)
Multiply it with the original excess .
- 18 * - 6 = 108
Way of writing = / 108 / - 216
Step 3 : -
Add new excess to the base
i.e.
100 - 18 = 82
and place it to the left of obtained digit .
Way of writing = 82 / 108 / -216
Step 4 : -
(82 + 1) / (08 - 3) / (100 - 16)
(08 - 3) is because -2 is for carry and 1(which converts into 100 going to right hand side is for - 16 .
Hence the answer is
830584
Example 1 : Find the cube of 104 .
Answer : -
Step 1 : -
In the above excess is 4 .
Find cube of 4 and retain two right most digit for answer . Remaining digits are for carry .
i.e.
43 = 64
Obtained digit = 64 Carry = 0
Step 2 : -
Double the previous excess . Find the new excess and multiply it with the previous excess .
i.e.
2 * 4 = 8
8 + 104 = 112 (Excess is 12)
Multiply this with the previous excess 4
12 * 4 = 48
Retain two right most digit for answer . Remaining digits are for carry .
Obtained digit = 48 64 Carry = 0
Step 3 : -
Now place the digit plus the double of excess to the left .
i.e.
112 4864
One Line representation
Way of writing = 112 / 048 / 064
Hence the cube of 104 is
1124864
Example 2 : - Find the cube of 106 .
Answer : -
Step 1 : -
Find the cube of excess .
63 = 216
Obtained digit = 16 Carry = 2
Step 2 : -
Double the excess . Add it to the original dgit . And multiply the new excess with the previous excess .
6 * 2 = 12 + 106 = 118 (New Excess = 18)
18 * 6 = 108
Add previous carry = 108 + 2 = 110
Obtained digit = 1016 Carry = 1
Step 3 : -
Add new excess(obtained in previous step) to the base , Add carry and then place it to the left of above obtained digit .
118 + 1 = 119
Obtained digit = 1191016
Hence the answer is 1191016 .
One Line representation of above question .
118 / 108 / 216
= 1191016
Example 3 : - Find the cubve of 94 .
Answer : -
Step 1 : -
Excess is - 6 .
Cube of -6 = - 216
Way of writing = / - 216
Step 2 : -
Double the excess . Add it to the original digit .
- 6 * 2 = - 12
94 - 12 = 82 (New excess is - 18)
Multiply it with the original excess .
- 18 * - 6 = 108
Way of writing = / 108 / - 216
Step 3 : -
Add new excess to the base
i.e.
100 - 18 = 82
and place it to the left of obtained digit .
Way of writing = 82 / 108 / -216
Step 4 : -
(82 + 1) / (08 - 3) / (100 - 16)
(08 - 3) is because -2 is for carry and 1(which converts into 100 going to right hand side is for - 16 .
Hence the answer is
830584
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