A Cube B Cube Formula
a cube b cube formula can be either a cube plus b cube formula or a cube minus b cube formula. These formulas are used to solve numerical expressions in mathematics. It comes in handy to find the factorial terms. It can also be used, when the given two numbers are of high values.
A Cube Plus B Cube Formula
The formula of a cube plus b cube is defined by,
a3 + b3 = (a + b)(a2 – ab + b2)
This formula is used to find the sum of the cube of any two given numbers.
a cube plus b cube formula derivation:
As we know,
(a + b)³ = a³ + 3ab(a + b) + b³
Rewritten as,
a³ + b³ = (a + b)³ – 3ab(a + b)
a³ + b³ = (a + b)[(a + b)² – 3ab]
a³ + b³ = (a + b)(a² + 2ab + b² – 3ab)
a³ + b³ = (a + b)(a² – ab + b²).
A Cube minus B Cube Formula
The formula of a cube plus b cube is defined by,
a3 – b3 = (a – b)(a2 + ab + b2)
This formula is used to find the difference of the cube of any two given numbers.
A Cube minus B Cube Formula Derivation
As we know,
(a – b)³ = a³ + 3ab(a – b) – b³
Rewritten as,
a³ – b³ = (a – b)³ + 3ab(a – b)
a³ – b³ = (a – b)[(a – b)² + 3ab]
a³ – b³ = (a – b)(a² – 2ab + b² + 3ab)
a³ – b³ = (a – b)(a² + ab + b²).
Solved Examples
Example 1:
Simplify: 543 + 103
Solution:
543 + 103,
Here a = 54 and b = 10,
By using the formula,
a3 + b3 = (a + b)(a2 – ab + b2),
543 + 103 = (54 + 10)[(54)2 – (54)(10) + (10)3]
= 64(2916 – 540 + 1000)
= 64(3376)
= 216064
Example 2:
Simplify: 93 – 53
Solution:
93 – 53,
Here a = 9 and b = 5,
By using the formula,
a3 – b3 = (a – b)(a2 + ab + b2),
93 – 53 = (9 – 5)[(9)2 + (9)(5) + (5)2]
= 4(81 – 45 + 25)
= 4(61) = 244
