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Home >> Standard Identities & their applications >> (a  b)^{3} = a^{3}  b^{3}  3ab(a  b) >> (A  B) Cube
Before you understand (a  b)^{3} = a^{3}  b^{3}  3ab(a  b), you are advised to read:
How to multiply Variables ?
How to multiply Constant and Variable ?
Multiplication of Polynomials ?
What is Exponential Form and Laws of Exponents ?
How identity of (a  b)^{3} = a^{3}  b^{3}  3ab(a  b) is obtained
Taking LHS of the identity:
(a  b)^{3}
This can also be written as:
= (a  b) (a  b) (a  b)
Multiply first two binomials as shown below:
= { a(a  b)  b(a  b) } (a  b)
= { a^{2}  ab  ab + b^{2} } (a  b)
Rearrange the terms in curly braces and we get:
= { a^{2} + b^{2}  ab  ab } (a  b)
Add above like terms, highlighted in red and we get:
= { a^{2} + b^{2}  2ab } (a  b)
Multiply trinomial with binomial as shown below:
= a^{2}(a  b) + b^{2}(a  b)  2ab(a  b)
= a^{3}  a^{2}b + ab^{2}  b^{3}  2a^{2}b + 2ab^{2}
Rearrange the terms and we get:
= a^{3}  b^{3}  a^{2}b  2a^{2}b + ab^{2} + 2ab^{2}
Add like terms, highlighted in green & red and we get:
= a^{3}  b^{3}  3a^{2}b + 3ab^{2}
Or we can further solve it:
Take 3ab common from the above blue highlighted terms and we get:
= a^{3}  b^{3}  3ab(a  b)
Hence, in this way we obtain the identity i.e. (a  b)^{3} = a^{3}  b^{3}  3ab(a  b) = a^{3}  b^{3}  3a^{2}b + 3ab^{2}
Let's try some example of this identity
Example 1: Solve (2a  3b)^{3}
Solution: This proceeds as:
Given polynomial (2a  3b)^{3} represents identity i.e. (a  b)^{3}
Where a = 2a and b = 3b
Now apply values of a and b on the identity i.e. (a  b)^{3} = a^{3}  b^{3}  3ab(a  b) and we get:
(2a  3b)^{3} = (2a)^{3}  (3b)^{3}  3(2a) (3b)(2a  3b)
Expand the exponential forms and we get:
= 8a^{3}  27b^{3}  3(2a)(3b)(2a  3b)
Solve multiplication process and we get:
= 8a^{3}  27b^{3}  18ab(2a  3b)
Hence, (2a  3b)^{3} = 8a^{3}  27b^{3}  18ab(2a  3b)
Example 2: Solve (5x  6y)^{3}
Solution: This proceeds as:
Given polynomial (5x  6y)^{3} represents identity i.e. (a  b)^{3}
Where a = 5x and b = 6y
Now apply values of a and b on the identity i.e. (a  b)^{3} = a^{3}  b^{3}  3a^{2}b + 3ab^{2} and we get:
(5x  6y)^{3} = (5x)^{3}  (6y)^{3}  3(5x)^{2}6y + 35x(6y)^{2}
Expand the exponential forms and we get:
= 125x^{3}  216y^{3}  3(25x^{2})(6y) + 3(5x)(36y^{2})
Solve multiplication process and we get:
= 8a^{3}  27b^{3}  450x^{2}y + 540xy^{2}
Hence, (5x + 6y)^{3} = 8a^{3}  27b^{3}  450x^{2}y + 540xy^{2}


