Question

Factorise \[{{a}^{4}}-{{b}^{4}}\]?

Answer 1

Hint: From the question we have been asked to factorise the given expression. For solving this question we will use the concept of algebra and its formulae. We will use the formula of difference of two squares that is \[{{a}^{2}}-{{b}^{2}}=\left( a-b \right)\left( a+b \right)\]. After using this difference of two squares formulae we will simplify the expression and find the answer.

Complete step by step answer:
Firstly, we can observe that the given expression \[{{a}^{4}}-{{b}^{4}}\] is of power four.
In algebra concept we have the formula of difference of two squares and not for power four which is as follows.
\[\Rightarrow {{a}^{2}}-{{b}^{2}}=\left( a-b \right)\left( a+b \right)\]
So, now we will simplify the given expression of power four and try to reduce it in the form of power two that is we bring it in the form of squares.
We can see that in the given expression we can rewrite the terms as \[{{a}^{4}}={{({{a}^{2}})}^{2}}\] and \[{{b}^{4}}={{({{b}^{2}})}^{2}}\].
So, we will write these terms in the place of the given terms in the expression.
So, the expression after rewriting will be reduced as follows.
\[\Rightarrow {{a}^{4}}-{{b}^{4}}\]
\[\Rightarrow {{\left( {{a}^{2}} \right)}^{2}}-{{\left( {{b}^{2}} \right)}^{2}}\]
So, here now we will use the formulae of difference of squares. So, the expression will be reduced as follows.
\[\Rightarrow \left( {{a}^{2}}+{{b}^{2}} \right)\left( {{a}^{2}}-{{b}^{2}} \right)\]
We can further reduce the expression by applying the formula of difference of square to the second term.
\[\Rightarrow \left( {{a}^{2}}+{{b}^{2}} \right)\left( {{a}^{2}}-{{b}^{2}} \right)\]
\[\Rightarrow \left( {{a}^{2}}+{{b}^{2}} \right)\left( a+b \right)\left( a-b \right)\]

Note: Students should be very careful in doing the calculations. Students should know the formulae of difference of squares which is \[\Rightarrow {{a}^{2}}-{{b}^{2}}=\left( a-b \right)\left( a+b \right)\]. If we stop our solution in this step \[\Rightarrow \left( {{a}^{2}}+{{b}^{2}} \right)\left( {{a}^{2}}-{{b}^{2}} \right)\] without further reduced it into \[\Rightarrow \left( {{a}^{2}}+{{b}^{2}} \right)\left( a+b \right)\left( a-b \right)\] then our solution is not complete.