## 1. Cube Numbers

Cubes and Cube Roots

Cube Numbers

Cube number

a cube is a 3-dimensional figure.  When we multiply a number by itself and then by itself again (thrice), the product is a cube number. It is also called as a perfect cube. That is, if a is a number, its cube is represented by a ^3.

A cube is a solid figure, which has all sides of equal length. The following table consist of cube numbers of the first ten numbers. Properties of cube numbers

1. The cube of a positive number is always positive.

Example43=4×4×4=64

2. The cube of a negative number is always negative.

Example: (−4)3=(−4)×(−4)×(−4)=−64

3. The cube of every even number is even.

Example23=8, 43=64, 63=216, 83=512, ...

Here, 8, 64, 216 and 512 are all even numbers.

4. The cube of every odd number is odd.

Example: 13=1, 33=27, 53=125, 73=343, ...

Here, 1, 27, 2125 and 343 are all odd numbers.

5. If a natural number ends at 0, 1, 4, 5, 6 or 9, its cube also ends with the same 0, 1, 4, 5, 6 or 9, respectively.

Example: (i) 10=1000

6. If a natural number ends at 2 or 8, its cube ends at 8 or 2, respectively.

Example(i) 23=8–

(ii) 83=512–

7. If a natural number ends at 3 or 7, its cube ends at 7 or 3, respectively.

Example: (i) 33=27

(ii) 73=343

8. A perfect cube does not end with two zeroes.

Example: 103=1000, 203=8000, …

9. The sum of the cubes of first n natural numbers is equal to the square of their sum.

That is, 13+23+33+43+….+n3=(1+2+3+4+…+n)2

Example: 13+23+33=1+8+27=36

(1+2+3)2=62=36

So, 13+23+33=(1+2+3)2

10. Each prime factor of a number appears three times in its cube.

Example:  63=216

Prime factor of 6 = 2×3

Prime factor of 216 = (2×2×2)×(3×3×3)

11. There are only three numbers whose cube is equal to itself.

(i) 03=0×0×0=0

(ii) 13=1×1×1=1

(iii) (−1)3=(−1)×(−1)×(−1)=−1

Example problems for cube numbers

1. Find the cube of 21.

Solution:

If you multiply a number by itself and then by itself again (thrice), the result is a cube number.

21=21×21×21=9261

Therefore, the cube of 21 is 9261.

2. Is 243 is a perfect cube? If not, find the smallest number to divide 243 to make it as a perfect cube.

Solution:

243=(3×3×3)×3×3

Here, factor 3×3=9 is leftover while grouping.

So, 243 is not a perfect cube.

To make it as a perfect cube, divide the given number by the leftover factor 9.

2439=27

Therefore, 9 is the smallest number to divide 243, and the obtained perfect square is 27.

## 2. Cube Root

Cube Root

Cube root

The inverse operation of a cube is cube root. The symbol used to represent the cube root is √3.

A cube root is a unique value that gives us the original number when we multiply itself by three times.

The cube root of a is denoted by 3√a or a1/3.

Example:

Find the cube root of 64.

Solution:  3√64=3√4×4×4−−−−−−−3√43 = 4

Therefore, the cube root of 64 is 4. The cube of 4 is 64.

The cube root of 64 is 4.

Cube root through prime factorisation

Steps to find the cube root of a number through prime factorisation:

Step 1: Find the prime factorisation of the given number.

Step 2: Group the factors in pair of three numbers (triplet).

Step 3: If there are no factor leftover, then the given number is a perfect cube. Otherwise, it is not a perfect cube.

Step 4: Now, take one factor common from each pair and multiply them.

Step 5: The obtained product is a cube root of a given number.

1. Find the value of  3√216.

Solution:

Let us first find the prime factor of 216.

Group the factors in pair of three numbers.

216=(2×2×2)×(3×3×3)

Here, no factor is leftover. Therefore, 216 is a perfect cube.

Now, take one factor common from each pair and multiply them.

3√216=2×3=6

Cube root of a cube number

Step 1: Choose any cube number.

Step 2: Start by forming groups of three digits from the number's right most digit. We get more than one group. Step 3: The first group will give you the ones (or unit) digit of the required cube root. You can calculate this using the ending digit of the first group.

Step 4: Now, take the second group and find which two cube numbers this group lies. Choose the smallest cube number and find the cube root of that number. This is the tens place of the required cube root.

Step 5: Join the numbers we get in the first group and second group. This is the required cube root of the given number.

Find the cube root of the cube number 79507.

Solution:

Step 1: The number 79507 is a cube number.

Step 2: Now, split this number into two groups. Step 3: Let us take the first group, 507. We know that "if a cube ends at 7, then its cube root ends at 3".

So, the ones place of the required number is 3.

Step 4: The second group is 79.

64<79<125

43<79<53

Here, the smallest cube number is 64, and the cube root of 64 is 4.

So, the tens place of the required cube root is 4.

Step 5: The required cube root is 43.