a + b + c Whole Cube Solved Examples
Que 1: Prove the correctness of the formula a + b + c whole cube by taking a= 1, b = 2, and c = 3.
Ans 1: 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
Que 2: Find the value of \((12)^3\) using the algebraic identity.
Ans 2: 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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