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Solve the algebraic expression ${(x + y)^3} $.

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Last updated date: 30th May 2024
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Answer
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Hint: In order to this question, to find the final value of the given algebraic expression ${(x + y)^3}$ , we will apply the algebraic formula, ${(a + b)^3} = {a^3} + {b^3} + 3{a^2}b + 3a{b^2}$ or we can first split the expression and solve by ${(a + b)^2} = {a^2} + {b^2} + 2ab$ .

Complete step by step solution:
Given algebraic expression is ${(x + y)^3}$ .

We can solve the expression by the help of cube of binomial process:
Step-1: First write the cube of the binomial \[{(x + y)^3} = (x + y) \times (x + y) \times (x + y)\]
Step-2: Multiply the first two binomials and keep the third one as it is
\[
  {(x + y)^3} = (x + y) \times (x + y) \times (x + y) \\
   \Rightarrow {(x + y)^3} = [x(x + y) + y(x + y)](x + y) \\
   \Rightarrow {(x + y)^3} = [{x^2} + xy + xy + {y^2}](x + y) \\
   \Rightarrow {(x + y)^3} = [{x^2} + 2xy + {y^2}](x + y) \\
\]
Step 3: Multiply the remaining binomial to the trinomial so obtained:
\[
  {(x + y)^3} = [{x^2} + 2xy + {y^2}](x + y) \\
   \Rightarrow {(x + y)^3} = x({x^2} + 2xy + {y^2}) + y({x^2} + 2xy + {y^2}) \\
   \Rightarrow {(x + y)^3} = {x^3} + 2{x^2}y + x{y^2} + {x^2}y + 2x{y^2} + {y^3} \\
   \Rightarrow {(x + y)^3} = {x^3} + 2{x^2}y + {x^2}y + x{y^2} + 2x{y^2} + {y^3} \\
   \Rightarrow {(x + y)^3} = {x^3} + 3{x^2}y + 3x{y^2} + {y^3} \\
   \Rightarrow {(x + y)^3} = {x^3} + {y^3} + 3{x^2}y + 3x{y^2} \\
   \Rightarrow {(x + y)^3} = {x^3} + {y^3} + 3xy(x + y) \\
\]
Note:
Alternative approach:
We can solve the given expression by the help of algebraic formula-
${(a + b)^3} = {a^3} + {b^3} + 3{a^2}b + 3a{b^2}$
Or by splitting the expression in the simplest form first.
Both methods will acquire the same result.
So, we have-
$
  {(x + y)^3} \\
   = (x + y){(x + y)^2} \\
   = (x + y)({x^2} + {y^2} + 2xy) \\
   = {x^3} + x{y^2} + 2{x^2}y + {x^2}y + {y^3} + 2x{y^2} \\
   = {x^3} + 3x{y^2} + 3{x^2}y + {y^3} \\
$
Hence, ${(x + y)^3} = {x^3} + 3x{y^2} + 3{x^2}y + {y^3}$.
An algebraic formula is a mathematical or algebraic law written as an equation. It's a two-sided equation with algebraic expressions on both sides. The algebraic formula is a simple, easy-to-remember formula for solving complex algebraic problems.