a3 - b3 Formula

Proof of a3-b3 Formula

The a3 - b3 formula is an algebraic identity which is read as a cube minus b cube. 

a3 - b3 Formula

Basic:

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

a3 - b3 Formula
a3 - b3 Formula
a3 - b3 Formula
a3 - b3 Formula

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Frequently Asked Questions on a3 - b3 Formula

a3 - b3 Formula in Algebra is an important formula of maths and is used in many questions and numerical,

a3 - b3 formula is expressed as.

a3 - b3 = (a - b) (a2 + ab + b2)