Properties of Cube
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Some properties of cubes of natural numbers are:-
Property 1: Cubes of all even numbers are even.
Example: 6
^{3} = 6 x 6 x 6 = 216
6 and 216 both are even numbers.
Property 2: Cubes of all odd natural numbers are odd.
Example: 7
^{3} = 7 x 7 x 7 = 343
7 and 343 both are natural odd numbers.
Property 3: The sum of the cubes of first n natural numbers is equal to the square of their sum.
1
^{3} + 2
^{3} + 3
^{3} + … + n
^{3} = (1 + 2 + 3 + … + n)
^{2}
Example: 1
^{3} + 2
^{3} + 3
^{3} + 4
^{3} = (1 + 2 + 3 + 4)
^{3}
1 + 8 + 27 + 64 = (10)
^{2}
100 = 100
Property 4 : Cubes of the numbers ending in digit 1, 4, 5, 6 and 9 are the numbers ending in the same digit.
Numbers |
Cubes |
1 |
1 |
2 |
8 |
3 |
27 |
4 |
64 |
5 |
125 |
6 |
216 |
7 |
343 |
8 |
512 |
9 |
729 |
10 |
1000 |
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Practice
1) Express 8
^{3} as sum of odd natural numbers.
2) Find the odd numbers to be added to get the following cubes.
a) 7
^{3}.
b) 12
^{3}
c) 4
^{3}
3) State the unit digits of the cubes of the following numbers.
a) (387)
^{3}
b) (83)
^{3}
c) (-81)
^{3}
d) (55)
^{3}
e) (680)
^{3}
Cube and Cube Roots
• Cube of Numbers
• Perfect Cube
• Properties of Cube
• Cube by Column method
• Cube of Negative numbers
• Cube of Rational numbers
• Cube-Root
• Finding cube-root by Prime Factorization
• Cube root of Rational numbers
• Estimating cube root
From Properties of cube to Exponents
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