What is the Formula of a^3-b^3? | A Cube Minus B Cube Formula

The a^3 – b^3 formula is (a – b) (a^2 + ab + b^2). This formula is used to calculate the difference of two cubes.

What is a^3-b^3 Formula?

The a³-b³ formula is (a – b) (a² + ab + b²)

Prove a Cube Minus b Cube Formula (a^3-b^3)

Let’s prove a cube minus b cube formula. To prove that a³ – b³ = (a – b) (a² + ab + b²).

LHS = RHS

LHS = a³ – b³

On Solving RHS side we get,

= (a – b) (a² + ab + b²)

On multiplying a and b separately with (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

Hence proved, LHS = RHS

Prove that (a + b + c) ^ 3 – a ^ 3 – b ^ 3 – c ^ 3 = 3(a + b)(b + c)(c + a)using a^3-b^3 Formula?

To prove (a+b+c)^3 −a^3 −b^3 −c^3 =3(a+b)(b+c)(c+a) using the a^3 − b^3 formula, let’s first expand (a+b+c)^3 using the binomial theorem:

(a+b+c)³=(a+b+c)(a+b+c)(a+b+c)

Now, applying the distributive property and multiplying each term

(a+b+c)³=(a²+ab+ac+ab+b²+bc+ac+bc+c²)(a+b+c)

=(a2+2ab+2ac+b2+2bc+c2)(a+b+c)

=a3+a2b+a2c+2a2b+2ab2+2abc+2a2c+2abc+2bc2+ab2+b3+b2c+a2c+2abc+ac2+bc2+c3

=a3+b3+c3+3(a2b+a2c+ab2+abc+ac2+bc2)

Now, we subtract a^3 +b^3 +c^3 from both sides:

(a+b+c)3−a3−b3−c3=3(a2b+a2c+ab2+abc+ac2+bc2)

Divide both sides by 3:

(a+b+c)³−a³−b³−c³/3 ​=a2b+a2c+ab2+abc+ac2+bc2

a2b+a2c+ab2+abc+ac2+bc2=ab(a+b)+ac(a+c)+bc(b+c)

Now, the equation becomes:

(a+b+c)³−a³−b³−c^3​/3=ab(a+b)+ac(a+c)+bc(b+c)

Multiply both sides by 3:

(a+b+c)³−a³−b³−c³=3(ab(a+b)+ac(a+c)+bc(b+c))

3(ab(a+b)+ac(a+c)+bc(b+c))=3(a+b)(a+c)(b+c)

Thus, we have proven that:

(a+b+c)³−a³−b³−c³=3(a+b)(a+c)(b+c)

Examples on a^3-b^3 Formula

Here are a some examples on a³-b³.

Example 1: Find the Value of 83³-53³ Using the a^3-b^3 Formula.

Solution:

To find 83³− 53³ using the a3−b³ formula:

Let us assume that a = 83 and b = 53

Apply a³−b³ formula: a³ – b³ = (a – b) (a² + ab + b²)

(83−53)(83²+(83×53)+53²)

=(83-53)(83²+4399+53²)

=(83-53)(6889+4399+2809)

=(30) (14097)

Now, multiply (a – b) by (a² + ab + b²)

= 422910

Therefore, 83³−53³=422910

Example 2: Simplify 19^3-20^3 Using a Cube Minus b Cube Formula

Solution

To find 19^3 – 20^3

Let us assume a = 19 and b = 20

Using formula a³ – b³ = (a – b) (a² + ab + b²)

a³ – b³ = (a – b) (a² + ab + b²)

19³ – 20³ = (19 – 20) (19² + (19)(20) + 20²)

= (-1) (361 + 380 + 400)

= (-1) (1141)

= -1141

Therefore, 19^3-20^3 = -1141

FAQ’s Related a^3-b^3 Formula

What is the Expansion of a³ – b³ Formula?

a³– b³ formula is read as a cube minus b cube. The a³– b³ expansion is expressed as (a – b) (a2 + ab + b2).

What is the Formula for a^3-b^3?

The formula for a^3 – b^3 is (a – b) (a^2 + ab + b^2)

What is a3 – b3 Formula in Algebra?

The formula for a3-b3 in algebra is a^3- b^3= (a – b) (a^2 + ab + b^2). This formula is used for factoring and simplifying the expressions involving the difference of two cubes.

What are a^3 – b^3 and a^3 + b^3 Formulas?

The formula of a³-b³ = (a – b) (a² + ab + b²)
The formula for a³+b³ = (a + b)(a² – ab + b²)

What is a Cube Minus b Cube Formula?

The formula of a Cube Minus b Cube is a3-b3 = (a – b) (a2 + ab + b2)