Class 8 Maths Chapter 7 Notes - Cubes and Cube Roots

Cube Numbers

The cube of a natural number n is denoted by n3, read as n-cube (or n to the power of 3) and is written as n × n × n. A number is a perfect cube if it is obtained by multiplying a natural number with itself three times.

For example, 27 is a perfect cube, because it is obtained by multiplying 3 with itself three times. 15 is also a natural number but it is not a perfect cube, because there is no natural number whose three times multiplication with itself gives us 15.

Interestingly, the cubes of even numbers are even whereas the cubes of odd numbers are odd.

Patterns in Cubes

• The difference of cubes of two consecutive numbers n and n + 1 can be written as: .
• If a number has n zeroes at the end, its cube will have 3n zeroes at the end.
• If a number ends with 0, 1, 4, 5, 6, or 9, its cube will also end with the same digit.

Let us see how prime factorisation method is used in identifying perfect cubes.

Prime Factorisation Method

Using prime factorisation method, we can check whether a number is a perfect cube or not. In this method, we divide the number by its smallest prime factor, then again we divide the new number with its smallest prime factor. We keep on doing the division until we reach to 1.

Afterwards, we write the number as the product of its prime factors and then group them in a set of three same factors. If all the prime factors can be grouped in a set of three, then the number is a perfect cube.

Learn how to find the factors of 32 using long division methnd.

Let’s check whether 2744 is a perfect cube or not.

The prime factors of 2744 are,

2744 = 2 × 2 × 2 × 7 × 7 × 7

On grouping the factors in a set of three same factors, we get,

2744 = (2 × 2 × 2) × (7 × 7 × 7)

Since, all the factors can be grouped in a set of three, 2744 is a perfect cube.

If a number is not a perfect cube, we can make it a perfect cube by multiplying or dividing it by some number. Let’s understand how we can do this for the number 108.

108 = 2 × 2 × 3 × 3 × 3

Now, make sets of three same factors, like this-

108 = 2 × 2 × (3 × 3 × 3)

The first group contains only two same factors (that is two 2’s). So, we would multiply both the sides with 2, so that the first group completes and 108 becomes a perfect cube.

108 × 2 = (2 × 2 × 2) × (3 × 3 × 3)
216 = 23 × 33 = (2 × 3)3
216 = 63

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NCERT Solutions for class 8 Maths Chapter 7
NCERT Solutions for class 8 Maths Chapter 7 Exercise 7.1
NCERT Solutions for class 8 Maths Chapter 7 Exercise 7.2

Cube Roots

The cube root of a number x is a number, which, when multiplied by itself three times, becomes x. The cube root of 64 is 4 because when 4 is multiplied by itself three times, becomes 64.

Let’s learn the methods to find the cube root of a number one by one.

Prime Factorisation

In this method, we will find the cube root of a number by breaking it into its prime factors and then we will make the sets of the 3 same factors.
Let us understand this with the help of an example.

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Find the cube root of 512.

Step 1: Find the prime factors of 512.

512 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2

Step 2: Make sets of the three same factors.

512 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2
512 = 22 × 22 × 22
512 = (2 × 2 × 2)2
512 = (8)2

Cube root of 512 = 8

Estimation Method

The cube root of large numbers can be found by the Estimation method. A number with unit place digit as 1, 8, 7, 4, 5, 6, 3, 2, 9 or 0 will have the unit place of its cube root as 1, 2, 3, 4, 5, 6, 7, 8, 9 or 0, respectively. This pattern is true for all cube numbers. This rule is used in estimation method.
Let’s find the cube root of 658503 using estimation method.

Step 1: Make groups of 3, starting from the right.

658 503

Step 2: Since, 503 ends with 3, its cube root will end with 7.

73 = 343

Therefore, the unit digit of cube root of 658503 = 7.

Step 3: Now, second group is 658.

83 = 512
93 = 729
83 < 658 < 93

The tens digit of the cube root will be the number whose cube is less than 658, i.e., 8.

Hence, 87 is the cube root of 658503.

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