Page 74 - Start Up Mathematics_8 (Non CCE)

4 Cubes and Cube Roots

We have studied about the solid figure cube which has 3 dimensions, length, breadth and height, that are
all equal. The space occupied by a cube is a × a × a, (where a is the length of a side). So, to find the space
occupied, we have to find the cube of its side.
In this chapter, we will learn about some special operations that will help us to find cubes and cube roots of
numbers.
Do you know?
Cube of a Number
The cube of a number is the number with 3 as its exponent raised There are just 4 numbers after 1
which are the sums of the cubes
to the power 3. of their digits.
3
3
Thus, if x is a number, then x is its cube where x = x × x × x. 153 = 1 + 5 + 3 3
3
3
3
3
For example, (i) 4 = (4 × 4 × 4) = 64, i.e., cube of 4 is 64. 370 = 3 + 7 + 0 3
3
3
3
3
(ii) 5 = (5 × 5 × 5) = 125, i.e., cube of 5 is 125. 371 = 3 + 7 + 1 3
3
3
407 = 4 + 0 + 7 3
Perfect Cube
Any number which is a product of three identical numbers is called a perfect cube.
3
3
3
For example, 1 = 1 × 1 × 1 = 1, 2 = 2 × 2 × 2 = 8, 3 = 3 × 3 × 3 = 27, etc.
So 1, 8, 27, 64, 125, etc. are perfect cubes.
Procedure for checking if a given natural number is a perfect cube
Step 1: Express the given natural number as a product of its prime factors.
Step 2: Group the factors in triplets of equal factors.
Step 3: If no factor is left after step 2, then the given natural number is a perfect cube, otherwise not.
To find a natural number whose cube is the given number, take out one factor from each triplet and multiply them.
Example 1: Which of the following are perfect cubes? 2 128

(a) 128 (b) 3,375 (NCERT) 2 64
Solution: (a) Writing 128 as a product of its prime factors, we get 2 32
128 = 2 × 2 × 2 × 2 × 2 × 2 × 2 2 16
2 8
Grouping them into groups of three, you can see that one 2 4
2 is left ungrouped. 2 2

So, 128 is not a perfect cube. 1
(b) 3,375 3 3,375
Writing 3,375 as a product of its prime factors, we get 3 1,125

3,375 = 3 × 3 × 3 × 5 × 5 × 5 3 375
Grouping them into groups of three, you can see that no 5 125
number is left ungrouped. 5 25
5 5
So, 3,375 is a perfect cube.
1