What is the Formula of a3 + b3 and a3−b3?

By Esha Dhawan|Updated : September 28th, 2022

Knowing the formula of a3−b3 and a3+b3 is mandatory for the candidates from the exam point of view as the maximum number of algebra questions asked in the competitive exams are based on it. The a3−b3 and a3+b3 formula is used to calculate the difference of two cubes. Let’s see how this formula is derived.

Formula of a3−b3

Let’s derive the formula for a3−b3 using the (a−b)3 formula

As we know, (a−b)3 = a3−b3 – 3ab(a – b)

Shift a3 and – b3 to the left-hand side and shift all other terms to the right-hand side to get:

a3−b3 = (a – b)3 + 3ab(a – b)

Taking common term (a-b) out we get following

a3−b3 = (a – b)[(a – b)2 + 3ab]

As we know (a - b)2 = (a2 - 2ab + b2)

So, we are replacing (a - b)2 in the above expression with (a2 - 2ab + b2). It can be written as:

a3−b3 = (a – b)[a2 - 2ab + b2 + 3ab]

On simplifying further, we get

a3−b3 = (a – b)(a2 + b2 + ab)

Therefore, the formula of a3−b3 is a3−b3 = (a−b)(a2 + b2 + ab)

What is the Formula of a3−b3?

This algebraic formula is used to calculate the difference of two cubes. It can easily be derived from using the (a −b)3 and its formula is a3−b3 = (a−b)(a2 + b2 + ab).

The formula of a3+b3 has been mentioned in the polynomial chapter in mathematics. Using these formulas complex calculations can be solved easily. To add the algebraic expressions, we can use the a3+b3 formula or the a3+b3+c3 formula. Similarly, for the subtraction of the algebraic expressions, the a3-b3 formula can be used.

The identities related to the a3+b3 formula have been shown in this post with examples or solutions to the questions based on these formulas.

Formula of a3+b3

The formula for a3 b3 for the different mathematical operations such as addition and subtraction are as follows:

  • The (a-b)3 formula = a3 – b3 – 3ab (a-b).
  • The formula of a3-b3 = (a2 + ab + b2)(a – b).
  • The (a + b)3 formula = a3 + b3 + 3ab(a + b)
  • The formula for a3+b3 = (a2 – ab + b2)(a + b).
  • The a3 + b3 + c3 formula = (a + b + c) (a2 + b2 + c2 – ab – bc – ca) + 3abc

Examples based on the a3b3 formula for addition and subtraction are given below:

Let a = 1, b =2, and c = 3. Now put these values in the above-mentioned formulas to get the answer.

  • The answer for the a3+b3 formula = (12 - 1x2 + 22)(1+2) = (1-2+4)(3) = 15.
  • The solution for (a + b)3 formula = 13 + 23 + 3x1x2(1+2) = 1+8+18 = 27.
  • The solution for a3 + b3 + c3 formula = (1+2+3)(12 + 22 + 32 - 1x2 - 2x3 - 3x1) = 18.

Steps to derive the a3+b3 formula

The step-by-step process to derive the formula has been explained below:

We know that (a + b)3 =(a + b) (a + b)2.

Then, (a + b)3 = (a + b)(a2 + 2ab + b2)

=a(a2 + 2ab + b2) + b(a2 + 2ab + b2)

= a3 + 2a2 b + ab2 + a2b + 2ab2 + b3

= a3 + 3a2 b + 3ab2 + b3

= a3 + b3 + 3ab(a + b)

From the above expression, (a + b)3 = a3 + b3 + 3ab(a + b). Now the formula of a3+b3 can be determined in the following ways:

  • (a + b)3 = a3 + b3 + 3ab(a + b)
  • a3 + b3 = 3ab(a + b) - (a + b)3

Thus, the formula of a3 + b3 = 3ab(a + b) - (a + b)3 or (a2 – ab + b2)(a + b).

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