Examples on Difference of Cubes
Example 1: Evaluate 123 – 83.
Solution:
Use the identity,
a3 – b3 = (a – b) (a2 + ab + b2)
where,
- a = 12
- b = 8
= 123 – 83
= (12 – 8) (122 + (12)(8) + 82)
= 4 (144 + 96 + 64)
= 4 (304)
= 1216
Example 2: Evaluate 153 – 103.
Solution:
Use the identity a3 – b3 = (a – b) (a2 + ab + b2)
where,
- a = 15
- b = 10
153 – 103
= (15 – 10) (152 + (15)(10) + 102)
= 5 (225 + 150 + 100)
= 5 (475)
= 2375
Example 3: Evaluate 193 – 93.
Solution:
Use the identity a3 – b3 = (a – b) (a2 + ab + b2)
where,
- a = 19
- b = 9
= 193 – 93
= (19 – 9) (192 + (19)(9) + 92)
= 10 (361 + 171 + 81)
= 5 (613)
= 3065
Example 4: Factorize x3 – 343.
Solution:
x3 – 343 = x3 – 73
Use the identity a3 – b3 = (a – b) (a2 + ab + b2)
where,
- a = x
- b = 7
= (x – 7) (x2 + (x)(7) + 72)
= (x – 7) (x2 + 7x + 49)
Example 5: Factorize y3 – 125.
Solution:
y3 – 125 = y3 – 53
Use the identity a3 – b3 = (a – b) (a2 + ab + b2)
where,
- a = y
- b = 5
= (y – 5) (y2 + (y)(5) + 52)
= (y – 5) (y2 + 5y + 25)
Example 6: Factorize x9 – 512.
Solution:
x9 – 512 = (x3)3 – 83
Use the identity a3 – b3 = (a – b) (a2 + ab + b2)
where,
- a = x3
- b = 8
= (x3 – 8) ((x3)2 + (x3)(8) + 82)
= (x3 – 23) (x6 + 8x3 + 64)
Again using the identity a3 – b3 = (a – b) (a2 + ab + b2)
where,
- a = x
- b = 2
= (x – 2) (x2 + (x)(2) + 22) (x6 + 8x3 + 64)
= (x – 2) (x2 + 2x + 4) (x6 + 8x3 + 64)
Difference of Cubes
Difference of Cubes is the formula in mathematics that is used to simplify the difference between two cubes. This formula is used to solve the difference of cubes without actually finding the cubes. This formula factorizes the difference of a cube and changes it into other forms. The difference of cube is also called the a3 b3 formula or the a3 – b3 formula.
In this article, we have covered the difference of cubes, the difference of cube formulas, various examples related to that formula, and others in detail.
Table of Content
- What is Difference of Cubes?
- Difference of Cubes Formula
- Derivation of Difference of Cube Formula
- Factoring Cubes Formula