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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. 

This formula helps students simplify complex algebraic expressions and solve equations quickly. It is commonly taught in middle and high school mathematics and is very useful in algebra, factorization, and competitive exams.

Uses of A Cube Minus B Cube Formula

The formula is mainly used for:

• Factorizing cubic expressions

• Simplifying algebraic calculations

• Solving polynomial equations

• Finding roots of algebraic expressions

• Speeding up calculations in exams

Whenever two perfect cubes are subtracted, this identity can be applied directly.

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

Conclusion

The A Cube Minus B Cube Formula is a simple yet powerful algebraic identity. By remembering this formula, students can solve factorization problems faster and more accurately. Regular practice of solved examples helps in understanding how to apply the identity correctly in different mathematical problems.