Question

Factorize the following:
\[{x^8} - {y^8}\]

Answer 1

Hint:First of all, write the given expression and simplify it by using the formula \[{a^2} - {b^2} = \left( {a - b} \right)\left( {a + b} \right)\]. Repeat the same method until we get the least simplification. So, use this method to reach the solution of the given problem.

Complete step-by-step answer:
Given expression is \[{x^8} - {y^8}\] which can be written as
\[ \Rightarrow {x^8} - {y^8} = {\left( {{x^4}} \right)^2} - {\left( {{y^4}} \right)^2}\]
We know that \[{a^2} - {b^2} = \left( {a - b} \right)\left( {a + b} \right)\]
By using this formula, we get
\[
   \Rightarrow {x^8} - {y^8} = {\left( {{x^4}} \right)^2} - {\left( {{y^4}} \right)^2} = \left( {{x^4} + {y^4}} \right)\left( {{x^4} - {y^4}} \right) \\
   \Rightarrow {x^8} - {y^8} = \left( {{x^4} + {y^4}} \right)\left( {{x^4} - {y^4}} \right) \\
\]
Now, simplifying \[{x^4} - {y^4}\] by using the formula \[{a^2} - {b^2} = \left( {a - b} \right)\left( {a + b} \right)\], we get
\[
   \Rightarrow {x^8} - {y^8} = \left( {{x^4} + {y^4}} \right)\left( {{x^4} - {y^4}} \right) \\
   \Rightarrow {x^8} - {y^8} = \left( {{x^4} + {y^4}} \right)\left[ {\left( {{x^2} - {y^2}} \right)\left( {{x^2} + {y^2}} \right)} \right] \\
\]
Now, simplifying \[{x^2} - {y^2}\] by using the formula \[{a^2} - {b^2} = \left( {a - b} \right)\left( {a + b} \right)\], we get
\[
   \Rightarrow {x^8} - {y^8} = \left( {{x^4} + {y^4}} \right)\left[ {\left( {{x^2} - {y^2}} \right)\left( {{x^2} + {y^2}} \right)} \right] \\
   \Rightarrow {x^8} - {y^8} = \left( {{x^4} + {y^4}} \right)\left[ {\left\{ {\left( {x - y} \right)\left( {x + y} \right)} \right\}\left( {{x^2} + {y^2}} \right)} \right] \\
  \therefore {x^8} - {y^8} = \left( {{x^4} + {y^4}} \right)\left( {{x^2} + {y^2}} \right)\left( {x + y} \right)\left( {x - y} \right) \\
\]
Thus, the factorization of \[{x^8} - {y^8}\] is \[\left( {{x^4} + {y^4}} \right)\left( {{x^2} + {y^2}} \right)\left( {x + y} \right)\left( {x - y} \right)\]

Note: We can verify our answer by doing the product of the obtained answer. If we get the same expression as given in the question our answer is correct otherwise wrong. We can use the direct formula \[{x^4} - {y^4} = \left( {{x^2} + {y^2}} \right)\left( {x + y} \right)\left( {x - y} \right)\] to solve the expre-Assion within no time.