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Chapter 6: Cubes and Cube Roots

    Introduction

    Cubes

    Cube Roots

    Exercise 6.1

    Exercise 6.2

    What Have We Discussed?

    Extra Curriculum Support

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Chapter 6: Cubes and Cube Roots > Cube Roots

Cube Roots

If the volume of a cube is 125 cm3, what would be the length of its side?

To get the length of the side of the cube, we need to know a number whose cube is 125.

Finding the square root, as you know, is the inverse operation of squaring.

Similarly, finding the cube root is the operation of finding cube.

We know that 23 =

So, we say that the cube root of 8 is .

We write 38 = 2. The symbol 3 denotes ‘cube-root.’

Consider the following:

StatementInference
13 = 131 = 1
23 = 838 = 323 = 2
33 = 27327 = 333 = 3
43 = 64364 =
53 = 125 3125 =
StatementInference
63 = 2163216 =
73 = 3433343 =
83 = 5123512 =
93 = 7293729 =
103 = 1000 31000 =

Consider 3375. Let's find its cube root using prime factorisation:

3375 = × × × × ×

= 33×53=3×53

Therefore, cube root of:

3375 = 33375 = × = (Enter factors in increasing order)

Similarly, to find 374088:

74088 = × × × × × × × × (Enter factors in increasing order)

= 23×33×73=2×3×73

Therefore,

374088 = × × =

Find the cube root of 8000

Prime factorisation of 8000 = × × × × × × × × (Enter factors in increasing order)

So,

38000 = 2 × 2 × 5 =

Find the cube root of 13824 by prime factorisation method.

13824 = × × × × × × × × × × × (Enter factors in increasing order)

= 23×23×23×33

Therefore,

313824= 2 × 2 × 2 × 3 =

Instructions

State true or false: for any integer m, m2 < m3. Why?

Let's consider four cases to check if the statement is true for all values. Case (i): m > 1 , Case (ii): m = 1 , Case (iii): m = 0 and Case (iv): m < 0.
Case (i) : m > 1. The statement is because m3 is than m2.
If m = 2, then 22 = 4 and 23 = 8 which gives us 4 8. Thus, m2 > m3.
Case (ii): m = 1. The statement is because m2 m3.
For m = 1, then 12 = 13.
Case (iii): m = 0. The statement is because m2 = m3.
For m = 0, then 02 = 03.
Case (iv): m < 0 (Negative Integers). The statement is because m3 m2.
If m = −2, then −22 = 4 and −23 = −8. So, 4 −8.
Thus, the statement is not true for all integers m.