Cubes and Cube Roots Class 8 Notes Maths Chapter 7

Students can go through AP 8th Class Maths Notes Chapter 7 Cubes and Cube Roots to understand and remember the concepts easily.

Class 8 Maths Chapter 7 Notes Cubes and Cube Roots

→ Numbers like 1729, 4104, 13832, are known as Hardy- Ramanujan Numbers. They can be expressed as sum of two cubes in two different ways.

→ Numbers obtained when a number is multiplied by itself three times are known as cube numbers. For example 1,8,27,… etc.

→ If in the prime factorisation of any number each factor appears three times, then the number is a perfect cube.

→ The symbol \(\sqrt[3]{ }\) denotes cube root. For example \(\sqrt[3]{27}\) = 3.

→ If we are able to express a number as a sum of two cubes of two positive integers then such number can be called as Hardy – Ramanujan
For example :
1. 4104 is Hardy – Ramanujan number because 4104 can be expressed as sum of 2³ and 16³
Check :
4104 = 2³ + 16³ [∵ 2³ = 8 ; 16³ = 16 × 16 × 16 = 256 × 16 = 4096]
= 8 + 4096 = 4104
So 4104 is sum of two cubes ; so it is Hardy – Ramanujan (or)
4104 = 9³ + 15³ [9³ = 9 × 9 × 9 = 729 15³ = 15 × 15 × 15 = 3375]
= 729 + 3375
= 4104
So 4104 can be expressed as either 2³ + 16³ (or) 9³ + 15³

Ex. 2. 13832 = 18³ + 20³ [18³ = 18 × 18 × 18 = 5832 = 20³ = 20 × 20 × 20 = 8000
= 5832 + 8000
= 13832
(or)
13832 = 2³ + 24³ [2³ = 2 × 2 × 2 = 8 24³ = 24 × 24 × 24 = 13824]
= 8 + 13824
= 13832
So we can call 13832 is a Hardy – Ramanujan number.
G.H. Hardy is British Mathematician Ramanujan is Indian Mathematician. Another example is 20683, which can be expressed as 27³ + 10³ (or) 24³ + 19³
Check :
20683 = 27³ + 10³ [∵ 27³ = 19683 ; 10³ = 1000]
= 19683 + 1000
= 20683
(or)
20683 = 24³ + 19³ [24³ = 13824 ; 19³ = 6859]
= 13824 + 6859
= 20683

Clarification : If 9 a Hardy – Ramanujan number
9³ = 1³ + 2³ but we cannot express 9 in another way of sum of two cubic numbers. Hence ‘9’ is not a Hardy – Ramanujan number.
H – Ramanujan numbers are also called as Taxi – cab number
Numbers that are the smelliest num-ber that can be expressed as sum of two cubes in ‘n’ distinict ways are described as Taxi – cab numbers.
Example :
1729 = 1³ + 12³ [1³ = 1, 12³ = 1728]
(or) = 9³ + 10³ [9³ = 729, 10³ = 1000]
Solid figures : Figures which have 3 – dimensions are known as solid figures.

1 unit solid ; bolded sides in the above figure are its 3 – dimensions.

Solid showing with 2 units on each side total 8 units = 2 × 2 × 2 = 2³
So this is a 2 unit cube have 8 cubic units

This a 3 units cube which have 3 × 3 × 3 = 27 units.
Perfect cubes or Cube numbers } A number which can be expressed as product of 3 same numbers.
Ex : 1, 8, 27, 64, 125 ………….. are cube numbers.
Why because
1 can be expressed as 1 × 1 × 1
8 can be expressed as 2 × 2 × 2
27 can be expressed as 3 × 3 × 3
64 can be expressed as 4 × 4 × 4
125 can be expressed as 5 × 5 × 5
First 4 cube numbers below 100 are 1, 8, 27, 64
First 9 cube numbers below 1000 are 1, 8, 27, 64, 125, 216, 343, 512, 729
and it is clear that we have 10 perfect cubes upto 1000.
Filling the table showing cubes of natural numbers 1 to 10.

Natural number(n)	1	2	3	4	5	6	7	8	9	10
Corresponding Cubic number n3	1	8	27	64	125	216	343	512	729	1000

Natural number(n) 1 2 3 4 5 6 7 8 9 10
Corresponding Cubic number n3 1 8 27 64 125 216 343 512 729 1000
From the above table we can conclude that cube of an even number is also even
Ex : (2 – 8) (4 – 64) (6 – 216) (8 – 512) …………
and cube of an odd number is also odd
Ex : (3 – 27) (5 -125) (7 – 343) (9 – 729)