Question

Factorize the following expression:
\[{x^6} - {y^6}\]

Answer 1

- Hint: First of all, write the given expression and simplify it by using the factorization formulas. Repeat the same method until we get the least simplification. So, use this concept to reach the solution of the given problem.

Complete step-by-step solution:

Given expression is \[{x^6} - {y^6}\] which can be written as
\[ \Rightarrow {x^6} - {y^6} = {\left( {{x^3}} \right)^2} - {\left( {{y^3}} \right)^2}\]
We know that \[{a^2} - {b^2} = \left( {a - b} \right)\left( {a + b} \right)\]
By using this formula, we get
\[
   \Rightarrow {x^6} - {y^6} = {\left( {{x^3}} \right)^2} - {\left( {{y^3}} \right)^2} = \left( {{x^3} + {y^3}} \right)\left( {{x^3} - {y^3}} \right) \\
   \Rightarrow {x^3} - {y^3} = \left( {{x^3} + {y^3}} \right)\left( {{x^3} - {y^3}} \right) \\
\]
Now, simplifying \[{x^3} - {y^3}\] by using the formula \[{a^3} - {b^3} = \left( {a - b} \right)\left( {{a^2} + ab + {b^2}} \right)\], we get
\[ \Rightarrow {x^6} - {y^6} = \left( {{x^3} + {y^3}} \right)\left( {x - y} \right)\left( {{x^2} + xy + {y^2}} \right)\]
Now, simplifying \[{x^3} + {y^3}\] by using the formula \[{a^3} + {b^3} = \left( {a - b} \right)\left( {{a^2} - ab + {b^2}} \right)\], we get
\[
   \Rightarrow {x^6} - {y^6} = \left( {x + y} \right)\left( {{x^2} - xy + {y^2}} \right)\left( {x - y} \right)\left( {{x^2} + xy + {y^2}} \right) \\
  \therefore {x^6} - {y^6} = \left( {x + y} \right)\left( {x - y} \right)\left( {{x^2} + xy + {y^2}} \right)\left( {{x^2} - xy + {y^2}} \right) \\
\]
Thus, the factorization of \[{x^6} - {y^6}\] is \[\left( {x + y} \right)\left( {x - y} \right)\left( {{x^2} + xy + {y^2}} \right)\left( {{x^2} - xy + {y^2}} \right)\]

Note: Factorising an expression is to write it as a product of its factors. To make sure an expression is fully factorised, we need to identify its highest common factor. Here we have used algebraic identities to solve this problem. Do up to the least possible simplification in these types of problems.