Answer
351.9k+ views
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 \[({a^3} + {b^3})\].
Complete step-by-step answer:
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.
Complete step-by-step answer:
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.
Recently Updated Pages
How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Why Are Noble Gases NonReactive class 11 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let X and Y be the sets of all positive divisors of class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x and y be 2 real numbers which satisfy the equations class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x 4log 2sqrt 9k 1 + 7 and y dfrac132log 2sqrt5 class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x22ax+b20 and x22bx+a20 be two equations Then the class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Trending doubts
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
At which age domestication of animals started A Neolithic class 11 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Which are the Top 10 Largest Countries of the World?
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Give 10 examples for herbs , shrubs , climbers , creepers
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference Between Plant Cell and Animal Cell
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Write a letter to the principal requesting him to grant class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Change the following sentences into negative and interrogative class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)