Cube and Cube Roots

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Cube Number

➤ Product we get when an number is multiplied three times by itself is called a cube number. or result we obtain when a number is cubed (raised to power of 3) is called a cube number.
2×2×2 = 8 , ⁴/₃×⁴/₃×⁴/₃ = 64/27 etc.
➤ When we multiply any integer three times by itself, cube number we get is called perfect cube.
So, a perfect cube is a cube of an integer.
(−2)×(−2)×(−2) = −8,
(−1)×(−1)×(−1) = −1,
1×1×1 = 1,
2×2×2 = 8,
3×3×3 = 27, etc
→ Cube of negative integer is negative.
→ Cube of positive integer is positive.

➤ For right now, we consider Cubes of only positive integers i.e, natural numbers,
1×1×1 =1,
2×2×2 = 8,
3×3×3 = 27, etc

Properties of Cube numbers

Number	Cube	Number	Cube
1 2 3 4 5 6 7 8 9 10	1 8 27 64 125 216 343 512 729 1000	11 12 13 14 15 16 17 18 19 20	1331 1728 2197 2744 3375 4096 4913 5832 6859 8000

→ The cubes of even numbers are even numbers and cubes of odd numbers are odd numbers.
8³ = 512, 16³ = 4096,
7³ = 343, 15³ = 3375,
→ When a cube number ends in 1, the number whose cube it is, will have 1 in unit’s place.
1³ = 1, 11³ = 1331
→ When a cube number ends in 8, the number whose cube it is, will have 2 in unit’s place.
2³ = 8, 12³ = 1728
→ When a cube number ends in 7, the number whose cube it is, will have 3 in unit’s place.
3³ = 27, 13³ = 2197
→ When a cube number ends in 4, the number whose cube it is, will have either 4 in unit’s place.
4³ = 64, 14³ = 2744
→ When a cube number ends in 5, the number whose cube it is, will have 5 in unit’s place.
5³ =125, 15³ = 3375
→ When a cube number ends in 6, the number whose cube it is, will have 6 in unit’s place.
6³ = 216, 16³ = 4096
→ When a cube number ends in 3, the number whose cube it is, will have 7 in unit’s place.
7³ = 343, 17³ = 4913
→ When a cube number ends in 2, the number whose cube it is, will have 8 in unit’s place.
8³ = 512, 18³ = 5832
→ When a cube number ends in 9, the number whose cube it is, will have either 9 in unit’s place.
9³ = 729, 19³ = 6859
→ When a cube number ends in 0, the number whose cube it is, will have 0 in unit’s place.
10³ = 1000, 20³ = 8000
→ Cube numbers can only have even number of zeros at the end which must be triple the number of zeros of the number whose cube it is.
10³ =1000, 20³ = 8000, 30³ =27000, 1000³ = 1000000000

Cube Roots

↪ Finding the cube root is the inverse operation of cubing.
    8 = 2×2×2, so we say that cube root of 8 is 2.
    27 = 3×3×3, so we say that cube root of 27 is 3.
→ The symbol ∛ denotes the cube root.
    ∛8 = 2, ∛27= 3

Statement	Inference	Statement	Inference
13 =1	∛1 = 1	63 =216	∛216 = 6
23 =8	∛8 = 2	73 =343	∛343 = 7
33 =27	∛27 = 3	83 =512	∛512 = 8
43 =64	∛64 = 4	93 =729	∛729 = 9
53 =125	∛125 = 5	103 =1000	∛1000 = 10

Finding Cube root

Prime Factorisation method

→ Write the given number as product of its prime factors
→ Group the prime factors in triples.
→ Express the tripled prime factors in exponential form of power of 3
→ Take the power of 3 as common
→ Put ∛ before the number in LHS and remove the power in RHS to get the required value.
e.g. 216 = 2×2×2×3×3×3
 ⇒ 216 = 2×2×2×3×3×3
 ⇒ 216 = 2³ × 3³
 ⇒ 216 = (2 × 3)³
 ⇒ ∛216 = 2×3 = 6
∴ ∛216 = 6

Square & Square Roots
	Irrational Numbers ➤