Cube roots of exact cubes.24 mathematics tricks and fun
Now what is a cube root? In simple words,Cube root of a number is a number which when multiplied three times,
by itself gives the number. Cube root of 8 is 2 and cube
of 2 is 8. Similarly cube root of 27 is 3 and cube of 3 is
27.
So to find the cube root quickly we should have a
working knowledge of the cubes of at least unit
digits. Now let us recollect them again.
1,. 1. 6. 216
2. 8. 7. 343
3. 27. 8. 512
4. 64. 9. 729
5. 125. 10. 1000
Now let us quickly brush up the steps in deriving the cube root..
1 group the digits in gps of three.if two or
one left , consider them as separate group.
2. Find the first and last digit of the cube
root
3. Digits 2,3,7,8 share their complements
from 10 as the last digit
4. Digits 1,4,5,6,9,0 have the same digit as
the last number
This is the algebraic explanation we will have to use in deriving the cuberoots of bigger numbers.
a is taken as the last digit here of the calculation
a3 3a2b 3 ab2 +3a2c. b3 + 3 abc +
3ac2
In finding the cube root of a 4 -6 digit number, since
only 2 digits in the answer, we will use only a3 and b3
When 7-9 digit number we will use a3 3 a2b. b3 and so
on
Here cube root of 1331 is shown.2 gps, cube of 1 is 1
and the number ends in 1,cube of 1 is 1, last digit is 1
so the answer is 11
Cube root of 2197 2 gps, cube of a digit which is
smaller than 2 is 1 , and the number ends in 7. so
cube of 3 ends in 7 so the cube root is 13
how to find the Cube root of 33 076 161. 3 gps,
number ends in 1 so the cube root of1 is 1.so the last
digit is 1, now what about first digit ?there are only 2 digits in this gp. Ie 33 we have to find a digit cube of
which is smaller than 33. . Cube of 3 is 27 so first digit
is 3
Now how to find the middle digit ?. First we will deduct
a3 fromthe whole number.now the number ends in 6.
3a2b we have to use ,ie 3×1×1b equals 3 b.now find a number which on multiplying with ,3 gives 6 ie 2.so the middle number is 2. we can confirm first number by proceeding further deduct 6as earlier.now 3ab + 3ac ie 3×1×2×2=12 and 3×1×cie 3c+ 12 deducting 12 gives 9
as end digit as earlier find a digit which on multiplying
with 3 gives 9 thus the answer is 321
We confirm using digital sum 321 ×321×321 = 18 ie 9 .digital sum of the number 33076161 is also 9.
Here we have two examples.cube of 405 244 so 2 gps
of 3 digits each.the first gp gp 405 so the cube of a
digit lesser than 405 is 7= 343 ,then last gp has the
end digit 4 .so cube 4 ends in 4 .so CR is 74
In this question 4657463 3 gps the first one is a single digit gp. So cube of a digit lesser than 4 is 1 . The last group ends in 3 so a cube which ends in 3 iss cube of
7 ie 343
Now middle digit.here we will subtract 343 from 463 gives1200.discard 0.the number now ends in 2.use
3a2b ie 3×7×7b=147b.with what digit we should
multiply 7 of 147 to get 2 as last digit ie 6 ,6×7 is 42
so middle digit is 6 and CR is 167.
In this there are 12 digits and 4gps of 3each. We will
get first digit. A digit the cube of which is less than
355is 7= 343.
Last gp ends in1 so invariably last digit of CR is 1since cube of 1 ends in1
Use 3a2b ie 3×1×1bie 3b..now deduct a cubed ie 1 from original number we get 355 045 312 44ie number now ends in 4 so with which digit we should multiply 3 to
get 4 as last digit.of course it is 8 ie 8×3=24 so b is 8,.
312 44-24 gives 3122
Next middle digit c . Use 3ab2+3a2c 3×1×8×8+ 3×1×1c192+3c on deducting 192 from 3122 gives 2930
so ince the digit ends in 0 c is 0 and CR is 7081
Thank you.in the t class we will study another
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