Question

Simplify ${{x}^{6}}-{{y}^{6}}$.

Answer 1

Hint: In the above question, we have been given an expression to simplify, which is written as ${{x}^{6}}-{{y}^{6}}$. For this, we can write the given expression as ${{\left( {{x}^{2}} \right)}^{3}}-{{\left( {{y}^{2}} \right)}^{3}}$ which is the difference of two cubes. For simplifying it, we have to use the algebraic identity given by ${{a}^{3}}-{{b}^{3}}=\left( a-b \right)\left( {{a}^{2}}+ab+{{b}^{2}} \right)$ and put $a={{x}^{2}}$ and $b={{y}^{2}}$ into the identity so that the given expression will get simplified as $\left( {{x}^{2}}-{{y}^{2}} \right)\left( {{x}^{4}}+{{x}^{2}}{{y}^{2}}+{{y}^{4}} \right)$. Then, we can further simplify the obtained expression by applying the algebraic identity given by ${{a}^{2}}-{{b}^{2}}=\left( a+b \right)\left( a-b \right)$ so that the given expression will be simplified completely.

Complete step by step answer:
Let us write the expression given in the above question as
$\Rightarrow E={{x}^{6}}-{{y}^{6}}$
Using the properties of the exponents, we can write the above expression as
$\Rightarrow E={{\left( {{x}^{2}} \right)}^{3}}-{{\left( {{y}^{2}} \right)}^{3}}$
Now, we know the algebraic identity given by ${{a}^{3}}-{{b}^{3}}=\left( a-b \right)\left( {{a}^{2}}+ab+{{b}^{2}} \right)$. On putting $a={{x}^{2}}$ and $b={{y}^{2}}$ into this identity, we can write the above expression as
$\begin{align}
  & \Rightarrow E=\left( {{x}^{2}}-{{y}^{2}} \right)\left( {{\left( {{x}^{2}} \right)}^{2}}+{{x}^{2}}{{y}^{2}}+{{\left( {{y}^{2}} \right)}^{2}} \right) \\
 & \Rightarrow E=\left( {{x}^{2}}-{{y}^{2}} \right)\left( {{x}^{4}}+{{x}^{2}}{{y}^{2}}+{{y}^{4}} \right) \\
\end{align}$
Now, we also know the algebraic identity given by ${{a}^{2}}-{{b}^{2}}=\left( a+b \right)\left( a-b \right)$. On putting $a=x$, and $b=y$ in this identity, we will get ${{x}^{2}}-{{y}^{2}}=\left( x+y \right)\left( x-y \right)$ so that we can write the above expression as
\[\Rightarrow E=\left( x+y \right)\left( x-y \right)\left( {{x}^{4}}+{{x}^{2}}{{y}^{2}}+{{y}^{4}} \right)\]

Hence, we have finally simplified the expression given to us in the question as \[\left( x+y \right)\left( x-y \right)\left( {{x}^{4}}+{{x}^{2}}{{y}^{2}}+{{y}^{4}} \right)\].

Note: We must remember all the algebraic identities in order to solve these types of questions. For simplifying the given expression, we can also write it as ${{\left( {{x}^{3}} \right)}^{2}}-{{\left( {{y}^{3}} \right)}^{2}}$. In this case, we have to use the algebraic identity given by ${{a}^{2}}-{{b}^{2}}=\left( a+b \right)\left( a-b \right)$ to simplify it as $\left( {{x}^{3}}+{{y}^{3}} \right)\left( {{x}^{3}}-{{y}^{3}} \right)$. Then we need to use the algebraic identities ${{a}^{3}}-{{b}^{3}}=\left( a-b \right)\left( {{a}^{2}}+ab+{{b}^{2}} \right)$ and ${{a}^{3}}+{{b}^{3}}=\left( a+b \right)\left( {{a}^{2}}-ab+{{b}^{2}} \right)$ to finally obtain the simplified expression.