Cube and Cube root

Cube 

If any number multiply by itself three time, then result is know as Cube .

We denote the Cube of x by x³

Example : 

x × x × x = x³

2×2×2 = 2³ = 8

Cube root

The is a given number x is the number whose Cube is x . 

We denote the Cube root of x by ∛x .

Example :

∛x×x×x = x

∛8 =∛2×2×2 = 2 

1) Rule :

Cube of that number which is made by 1 .

Example :

1) 11³

Step I       11×11×11

Step II  = 121×11→  1 / 1+2 / 2+1 / 1

Step III  =                   1 /3 /3 /1

                =                    1331

2) 111³

Step I       111×111×111

Step II  =  12321×111

Step III  =  1/ 1+2 / 1+2+3 / 2+3+2 / 3+2+1 / 2+1 / 1

Step IV   =  1/ 3/ 6 /7 /6 /3 /1

                =   1367631

3) 1111³+

Step I        1111×1111×1111

Step II  =    1234321×1111

Step III  =    1/ 1+2 / 1+2+3 / 1+2+3+4 / 2+3+4+3/ 3+4+3+2/ 4+3+2+1/ 3+2+1 / 2+1 / 1

Step IV   =     1 /3 /6 /10 /12 /12/ 10 /6 /3 /1

                 =     1371330631

2) Rule :

Cube of that number which is made by 2 .

Example :

 1) 22³

Step I       22×22×22

Step II  = 484×22 →  2(4) / 2(4+8) / 2(8+4) / 2(4)

Step III  =                   8 /24 /24 /8

Step IV   =                  10648

2) 222³

Step I       222×222×222

Step II  =  49284×222

Step III  =  2(4)/ 2(4+9) / 2(4+9+2) / 2(9+2+8) / 2(2+8+4) / 2(8+4) / 2(4)

Step IV   =  8/ 26/ 30 /38 /28 /24 /8

                =     10941048

3) 2222³

Step I        2222×2222×2222

Step II  =    4937284×2222

Step III  =    2(4)/ 2(4+9) / 2(4+9+3) / 2(4+9+3+7) / 2(9+3+7+2)/ 2(3+7+2+8)/ 2(7+2+8+4)/ 2(2+8+4) / 2(8+4) / 2(4)

Step IV   =     8/26/32/46/42/40/ 42/28/24/8

                 =    10970645048

3) Rule :

Similarly we can find Cube of 3 , 4 .......

4) Rule :

Cube Root by prime factorization method .

Example :

1)  ∛3375 = ?

3  | 3375

3  |  1125

3  |    375

5  |    125

5  |      25

    |        5

   ∛3375 =   ∛3×3×3×5×5×5 = 3×5 = 15

It is 15×15×15 = 3375

2)   ∛1728

2 | 1728

2 |   864

2 |   432

2 |   216

2 |   108

2 |     54

3 |     27

3 |       9

           3

  ∛1728 =   ∛2×2×2×2×2×2×3×3×3

              =   8×8×3 = 12

3) Find the cube root of 2744 .

Solution:

2 |2744

2 |1372

2 |  686

7 |  343

7 |    49

7 |      7

         1

∛2744    =   ∛2×2×2×7×7×7

              =   2×7 

              =   14

4) Is 53240 a perfect cube ? If not, then by which smallest natural number should 53240 be divided so that the quotient is a perfect cube ?

Solution:

53240 = 2×2×2×11×11×11×5

The prime factor 5 does not appear in a group of three. So, 53240 is not a perfect cube. In the factorisation 5 appears only one time. If we divide the number by 5, then the prime factorisation of the quotient will not contain 5 .

So,     53240 ÷ 5 = 2×2×2×11×11×11

Hence the smallest number by which 53240 should be divided to make it a perfect cube is 5.

The perfect cube in that case is = 10648.

5) Is 1188 a perfect cube ? If not, by which smallest natural number should 1188 be divided so that the quotient is a cube ?

Solution :

1188 = 2×2×3×3×3×11

The primes 2 and 11 do not appear in groups of three. So, 1188 is not a perfect cube. In the factorisation of 1188 the prime 2 appears only two times and the prime 11 appears once. So, if we divide 1188 by 2×2×11 = 44, then the prime factoriation of the quotient will not contain 2 and 11.

Hence the smallest natural number by which 1188 should be divided tomake it a perfect cube is 44.

And the resulting perfect cube is 1188÷44 = 27 = 3³

6) Is 392 a perfect cube ? If not, find the smallest natural number by which 392 must be multiplied so that the product is a perfect cube.

Solution.

392 =  2×2×2×7×7

The prime factor 7 does not appear in a group of three. Therefore, 392 is not a perfect cube. To make its a cube, we need one more 7. In that case 

392×7= 2×2×2×7×7×7 = 2744 Which is perfect cube.

Hence the smallest natural number by which 392 should be multiplied to make a perfect cube is 7.

7) Find the cube root of 857375 .

Solution :

Step 1 From groups of three starting from the rightmost digit of 857375.

857                          375

↓                              ↓

second group          first group

We can estimate the cube root of a given cube number through a step by step process. 

We get 375 and 857 as two groups of three digits each.

Step 2 First group, i.e., 375 will give you the one's digit of the required cube root.

The number 375 ends with 5. We know that 5 comes at the unit's place of a number, only when it's cube root ends in 5.

So, we get 5 at the unit's place of the cube root.

Step 3 Now take another group, i.e. , 857.

We know that  

9³= 729 and  

10³= 1000.

Also, 729 < 857 < 1000 . We take the one's place, of the smaller number 729 as the ten's place of the required cube root. So, we get ∛857375 = 95 .

Exercise

1) Which of the following numbers are not perfect cubes ?

a) 216    b) 128    c) 1000    d) 46656

2) (11)³ = ?

a) 1221    b) 1331    c) 3131    d) 3113 

3) (333) = ?

a) 369260937    b) 3691160937    c) 669336739    d) 3336333

Answer 

1) b    2) b    3) a