a3 - b3 Formula

The a³ - b³ formula is an algebraic identity which is read as a cube minus b cube. The a³ - b³ formula is used to find factorise cubes of binomials. The a³ - b³ formula helps us find the difference between cubes of two numbers easily without calculating the cubes of numbers. The a³ - b³ formula is applicable for all values of a and b. Sometimes it is also called ‘the difference of cubes’ formula. We will be discussing aspects of the a³ - b³ formula, along with solved examples, and understanding the identity involved. Get the List of Maths formulas. 

What Is a³ - b³ Formula?

The a³ - b³ = (a-b)(a²+ab+b²) which can verified as well as can be derived from the formula (a-b)³.

Lets discuss both aspects.

To verify a³ - b³ = (a-b)(a²+ab+b²) Solving the RHS

(a - b) (a² + ab + b²)

Using binomial multiply with trinomial

= 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 we have a³-b³

Therefore, LHS=RHS.

To prove by using (a-b)³ identity

Well we know that

(a - b)³ = a³ - b³ - 3ab(a - b)

So, we get

(a - b)³ +3ab(a - b) = a³ - b³

Taking (a - b) as common, we get

a³ - b³ = (a - b)((a -b)² + 3ab))

= (a - b)(a² + b² - 2ab + 3ab)

= (a - b)(a² + b²+ab)

Therefore the value of a³ - b³ is (a - b)(a + b² +ab)

Examples on a³ - b³ Formula

Example 1: Find the value of 1003 - 83 using the a³ - b³ formula.

Solution:

To find: 100³ - 8³.

Let us assume that a = 100 and b = 8.

We will substitute these in the formula of a³ - b³

a³ - b³ = (a - b) (a² + ab + b²)

100³-8³ = (100-8)(100² + (100)(8) + 8²)

= (92) (10000+800+64)

= (92)(10864)

=999488

Answer: 100³ - 8³ = 999488.

Example 2: Factorize 27x³ + 125y³

Solution: 27x³+125y³

= (3x)³ + (5y)³

= (3x+5y) (9x²-15xy + 25y²)

[: (a³ + b³) = (a + b) (a²-ab+b²)].

Answer: 27x³ + 125y³= (3x+5y) (9x²-15xy + 25y²)

Example 3: Factorise a³ + b³ +a+b

Solution: a³ + b³ +a+b

= (a³ + b³)+(a+b)

= (a + b) (a² − ab + b²) + (a+b)

[: (a³ + b³) = (a + b)(a²-ab+b²))

= (a + b) ×{(a²-ab+b²) +1}

= (a + b) (a²- ab + b² + 1).

Answer: (a³ + b³ +a+b) = (a + b) (a² − ab + b²+1).

Example 4: If x+y = 12 and xy = 27, find the value of (x ³ + y ³)

SOLUTION

We have

(x ³ + y ³) = (x + y)(x ² - xy + y ²)

= (x+y) [(x + y)²-3xy]

= 12×[(12)²-3 × 27]

= 12X (144-81) = 12 X 63 = 756.

Answer: (x ³ + y ³) = 756