How do you factorise the given equation ${x^6} - {y^6}$?

Hint: In this question we need to find the factor of algebraic expression ${x^6} - {y^6}$, but this expression has two variables $x$ and $y$. To factor this expression we will use some of the basic algebraic identities such as ${a^2} - {b^2} = (a + b)(a - b)$ and ${a^3} - {b^3} = (a - b)({a^2} + ab + {b^2})$. Also the knowledge of exponents is also required. Please try once yourself before looking at a complete solution.

Complete step by step answer:
Let us try to solve this in which we are asked to factor given algebraic expression ${x^6} - {y^6}$. To solve this type of question where we are asked the factor of algebraic expression knowledge of exponents is required for some manipulation. So that we apply known algebraic identities to factor it.

To solve this we use these algebraic identities.
$ \Rightarrow {a^2} - {b^2} = (a + b)(a - b)$
$ \Rightarrow {a^3} - {b^3} = (a - b)({a^2} + ab + {b^2})$
Expression ${x^6} - {y^6}$ can be also written as
$ \Rightarrow {x^6} - {y^6} = {({x^2})^3} - {({y^2})^3}$............................$eq(1)$
Because we know that from the law of exponents${({x^a})^b} = {x^{a \cdot b}}$.

Now applying the identity ${a^3} - {b^3} = (a - b)({a^2} + ab + {b^2})$ in $eq(1)$, we get
$ \Rightarrow {({x^2})^3} - {({y^2})^3} = ({x^2} - {y^2})({x^2} + {(xy)^4} + {y^4})$.................$eq(2)$
Now applying the identity ${a^2} - {b^2} = (a + b)(a - b)$ in $eq(2)$ , we get
$ \Rightarrow {({x^2})^3} - {({y^2})^3} = (x + y)(x - y)({x^2} + {(xy)^4} + {y^4})$
Hence the factor of given algebraic expression
$ \Rightarrow {x^6} - {y^6}$ is$(x + y)(x - y)({x^2} + {(xy)^4} + {y^4})$.

Note: We can also solve for the factor of expression by writing ${x^6} - {y^6} = {({x^3})^2} - {({y^3})^2}$ using above algebraic identities and ${a^3} + {b^3} = (a + b)({a^2} - ab + {b^2})$, we also get the same answer as above but this process is lengthy. After a series of cancellation, we reach the same answer from this method. You can try to solve by this method to check that you are getting the same factor or not. To solve these types of questions you are required to know the basic algebraic identities without them you would not be able to solve, so try to learn algebraic identities.