The difference of the cubes {of two numbers} formula is the a cube minus b cube formula. Without actually computing the cubes, the a3 – b3 formula is utilized to discover the difference between them. It can also be used to factorize cubic binomials. The sum of cubes {of two numbers} formula is represented as a3 – b3. Without having to calculate the cubes, the cube plus b cube formula is used to obtain the sum of the two cubes. It is also used in the factorization of cube binomials. In this section, we’ll go through the many aspects of the a3 – b3 formula. We may evaluate the a3 – b3 formula by multiplying {a – b} {a2 + ab + b2} and seeing if you obtain a3 – b3. The a3 – b3 formula.

Proof of a3 – b3 Formula

Let us see the proof of a cube plus b cube a3 – b3 formula below.

- To prove that a3 – b3 = {a – b} {a2 + ab + b2}

we need to prove here LHS = RHS. Let’s begin with the following steps.

LHS = a3 – b3

On Solving RHS side we get,

= {a – b} {a2 + ab + b2}

On multiplying the a and b separately with {a2 + ab + b2} we get

= 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

Now Let us see the proof of a cube plus b cube a3 – b3 formula below.

To prove that {a – b}3 = {a3 – b3 } + 3ab{a – b}

we need to prove here LHS = RHS. Let’s begin with the below steps.

{a – b}3 = {a3 – b3 } + 3ab{a – b}

Now, Subtract the 3ab {a – b} from each side of the above Equation.

{a – b}3 – 3ab{a – b} = a3 – b3

Therefore, the formula for {a3 – b3} is

{a3 – b3}= {a – b}3 – 3ab{a – b}

{a3 – b3}= {a – b}[{a – b}2 – 3ab]

{a3 – b3}= {a – b}[a2 – 2ab – b2 – 3ab]

{a3 – b3}= {a – b}*{a2 – ab – b2}

Therefore, the formula for {a3 – b3} is

{a3 – b3}= {a – b}*{a2 – ab – b2}

So,

{a – b} and {a2 – ab – b2}

are the factors of {a3 – b3}

Thus, {a – b}3 = {a3 – b3 }- 3ab{a – b}

How to Use a3-b3 Formula?

The following steps can be taken while using a3 – b3 formula.

- Firstly, observe the pattern of the 2 numbers and check whether the numbers have 3 as power or not.
- After that, note down the formula of and that is a3 – b3 = { a – b} { a 2 + a b + b2}
- Now, Substitute the values of the a and the b in a3 – b3 formula.
- Lastly, multiply a and b one by one with {a2 + ab + b2} and simplify to get the result.

Point To remember:

- {a – b}3 = {a3 – b3 } + 3ab{a – b}
- a3 – b3 = {a – b} {a2 + ab + b2}