We use many algebraic identities to solve mathematical problems in our daily lives. a + b + c whole cube is an algebraic identity that is used to find the value of a cube of sum of three numbers.

The expression for the value of a + b + c whole cube is given as:

(a+b+c)3=a3+b3+c3+3(a+b)(b+c)(c+a)

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In this article, we shall learn about this algebraic identity in detail along with its derivation and some solved examples for better understanding of the concept.

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a + b + c whole cube is used to find the cube of the sum of three real numbers. The formula can be used to factorize some special type of functions. The formula is written as:

(a+b+c)3=a3+b3+c3+3(a+b)(b+c)(c+a)

 Let us derive formula using simple mathematical operations:

Proof:

\((a + b + c)^3 = a^3 + b^3 + c^3 + 3 (a +b) (b + c) (a+ c)\)

It can be written as:

Consider the L.H.S of equation (1),

\( (a + b + c)^3 – a^3 – b^3 – c^3 \)

= \(a^3 + b^3 + c^3 + 3 ab (a + b) + 3 bc (b + c) + 3 ac (a + c) +6 abc – a^3 – b^3 – c^3\)

=\( 3 ab (a + b) + 3 bc (b + c) + 3 ac (a + c) +6 abc\)

= \(3 [ ab (a + b) + bc (b + c) + ac (a + c) + 2 abc] \)

= \(3 [ ab (a + b) + b^2c + bc^2 + abc + a^2c + ac^2 + abc ]\)

= \(3 [ ab (a + b) + (abc + b^2c) + (abc + a^2c) + (bc^2 + ac^2) ]\)

= \(3 [ ab (a + b) + bc (a + b) + ac (a + b) + c^2 (a + b) ]\)

= \(3 [ (a + b) (ab + bc + ac + c^2) ]\)

= \(3 [ (a + b) { (c^2 + bc) + ( ab + ac) } ]\)

= \(3 [ (a + b) { c ( b + c ) + a ( b + c ) } ]\)

= 3 (a + b) ( b + c) ( a + c )

which is equal to R.H.S of equation (1).

Hence Proved.

Summary

Let us summarize what we have learnt about a + b + c whole cube:

• a + b + c whole cube formula is used to find the cube of the sum of three real numbers.
• The formula or the algebraic identity to find the value of a + b + c whole cube is written as: \((a + b + c)^3 = a^3 + b^3 + c^3 + 3(a + b)(b + c)(c + a)\). 
• Here, a, b and c are real numbers.

a + b + c Whole Cube Solved Examples

Example 1: Prove the correctness of the formula a + b + c whole cube by taking a= 1, b = 2, and c = 3.

Solution: We know that the value of a + b + c whole cube is written as:

\((a + b + c)^3 = a^3 + b^3 + c^3 + 3(a + b)(b + c)(c + a)\)

Given that:

a = 1, b = 2 and c = 3. Let us substitute the values of a, b, and c in the above formula:

\((a + b + c)^3 = a^3 + b^3 + c^3 + 3(a + b)(b + c)(c + a)\)

We need to prove LHS = RHS,

LHS = \((a + b + c)^3 \)

LHS = \((1 + 2 + 3)^3\)

= (6)^3 = 216

RHS = \(a^3 + b^3 + c^3 + 3(a + b)(b + c)(c + a)\)

= \(1^3 + 2^3 + 3^3 + 3(1 + 2)(2 + 3)(3 + 1)\)

= \(1 + 8 + 27 + 3(3)(5)(4)\)

=\( 36 + 180\)

= 216

Hence Proved.

Example 2: Find the value of \((12)^3\) using the algebraic identity.

Solution: We know that:

\((a + b + c)^3 = a^3 + b^3 + c^3 + 3(a + b)(b + c)(c + a)\)

Let us write 12 as 3 + 4 + 5.

Using the identity:

\((3 + 4 + 5)^3 = 3^3 + 4^3 + 5^3 + 3(3 + 4)(4 + 5)(3 + 5)\)

\(12^3 = 27 + 64 + 125 + 3(7)(9)(8)\)

\(12^3 = 216 + 1512\)

\(12^3 = 1728\)

Therefore, \(12^3 = 1728\).

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