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# What is the formula of $({a^3} + {b^3})$ ?

Last updated date: 16th Sep 2024
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Hint: The question is related to the algebraic expression. In the algebra we have some standard algebraic identities for squares and cubes. So here we have to find the formula for $({a^3} + {b^3})$, first we determine ${(a + b)^3}$ and then we determine $({a^3} + {b^3})$.

The algebraic expression is a combination of the constants, variables and it includes the arithmetic operation symbols. In algebraic expression we have three different kinds namely, monomial, binomial and polynomial expressions.
Now we will consider the algebraic expression
$\Rightarrow (a + b)$
Now multiply the above term by $(a + b)$, the we have
$\Rightarrow (a + b)(a + b)$
On multiplying we have
$\Rightarrow a(a + b) + b(a + b)$
$\Rightarrow a.a + a.b + b.a + b.b$
When the same variable multiplied twice we write in the form of the exponent so we have
$\Rightarrow {a^2} + 2ab + {b^2}$
Therefore we have
$\Rightarrow {(a + b)^2} = {a^2} + 2ab + {b^2}$ ----- (1)
Now to the equation (1) again we multiply by $(a + b)$, we have
$\Rightarrow {(a + b)^2}(a + b) = (a + b)\left( {{a^2} + 2ab + {b^2}} \right)$
On multiplying we have
$\Rightarrow {(a + b)^3} = a({a^2} + 2ab + {b^2}) + b({a^2} + 2ab + {b^2})$
$\Rightarrow {(a + b)^3} = a.{a^2} + 2ab.a + a.{b^2} + b.{a^2} + 2ab.{b^2} + b.{b^2}$
When the same variable multiplied twice or more we write in the form of the exponent so we have
$\Rightarrow {(a + b)^3} = {a^3} + 2{a^2}b + a{b^2} + {a^2}b + 2a{b^2} + {b^3}$
On simplifying we have
$\Rightarrow {(a + b)^3} = {a^3} + 3ab(a + b) + {b^3}$
Therefore we have
$\Rightarrow {(a + b)^3} = {a^3} + {b^3} + 3ab(a + b)$ ----- (2)
Take $3ab(a + b)$ to LHS we get
$\Rightarrow {(a + b)^3} - 3ab(a + b) = {a^3} + {b^3}$
This can be written as
$\Rightarrow {a^3} + {b^3} = {(a + b)^3} - 3ab(a + b)$
Take $(a + b)$ as common in the RHS we have
$\Rightarrow {a^3} + {b^3} = (a + b)\left( {{{(a + b)}^2} - 3ab} \right)$---------- (3)
Substitute the (1) in (3) we have
$\Rightarrow {a^3} + {b^3} = (a + b)\left( {{a^2} + {b^2} + 2ab - 3ab} \right)$
On simplifying we have
$\Rightarrow {a^3} + {b^3} = (a + b)\left( {{a^2} + {b^2} - ab} \right)$
Hence we have determined the value of $({a^3} + {b^3})$
Therefore the formula of $({a^3} + {b^3})$ is $(a + b)\left( {{a^2} + {b^2} - ab} \right)$
So, the correct answer is “ $(a + b)\left( {{a^2} + {b^2} - ab} \right)$”.

Note: Since the question contains the algebraic expression and formula of the algebraic identities, it is easy to solve the problem if we knew the standard algebraic formulas. We should take care of signs because they may sometimes while solving. The like terms can be cancelled or added but not unlike terms.