The a cube – b cube formula, i.e. a3 – b3 formula, is also called the difference of cubes formula in maths. This formula is treated as one of the algebraic identities in maths. This formula is used to find the difference between the cubes of two numbers without actually calculating the cubes. The formula of a3 – b3 is also used to factorise the binomials in the form of cubes. In this article, you will learn the cube minus b cube formula along with proof and examples.
a^3 – b^3 Formula
The formula for a3 – b3 is expressed in terms of the multiplication of a binomial and trinomial. This is used to evaluate the numerical as well as polynomial expressions.
The formula of a3 – b3 is given by:
| a3 – b3 = (a – b)(a2 + ab + b2) |
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Let’s have a look at the proof of this formula.
A cube – B cube Formula Proof
To prove: a3 – b3 = (a – b)(a2 + ab + b2)
Consider LHS = a3 – b3
RHS = (a – b)(a2 + ab + b2)
The difference of numbers a3 and b3 formula can be verified by taking the RHS.
RHS = (a – b)(a2 + ab + b2)
= = a (a2 + ab + b2) – b(a2 + ab + b2)
= a3 + a2b + ab2 – a2b – ab2 – b3
= a3 + a2b – a2b + ab2– ab2 – b3
= a3 – 0 – 0 – b3
= a3 – b3
= LHS
Therefore, a3 – b3 = (a – b)(a2 + ab + b2)
Hence proved.
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Solved Examples
Example 1: Evaluate: 1033 – 27
Solution:
1033 – 27
= 1033 – 33
This is of the form a3 – b3.
Using the formula a3 – b3 = (a – b)(a2 + ab + b2),
1033 – 27 = 1033 – 33 = (103 – 3)[(103)2 + (103)(3) + (3)2]
= 100(10609 + 309 + 9)
= 100 (10927)
= 1092700
Therefore, 1033 – 27 = 1092700
Example 2: Factorise 64x3 – 216 using a cube minus b cube formula.
Solution:
64x3 – 216
= (4x)3 – (6)3
This is of the form a3 – b3.
Using the formula a3 – b3 = (a – b)(a2 + ab + b2),
64x3 – 216 = (4x – 6)[(4x)2 + (4x)(6) + (6)2]
= (4x – 6)(16x2 + 24x + 36)
Therefore, 64x3 – 216 = (4x – 6)(16x2 + 24x + 36)
Video Lesson on Algebraic Expansion
Practice Questions
- Simplify 303 – 153 using the a3 – b3 formula.
- Evaluate: 2003 – 125
- Factorise: 343x3 – 8y3
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