Question

Simplify the expression $ \dfrac{{{x^4} - {y^4}}}{{\left( {{x^4} + 2{x^2}{y^2} + {y^4}} \right)\left( {{x^2} - 2xy + {y^2}} \right)}} $ ?

Answer 1

Hint:
We are given an algebraic expression and we have to simplify it. To simplify an algebraic expression we can use algebraic identities. The expression which are given can be solved by using three different algebraic identities The Identity used here are:
1. Difference of the square of two numbers:
  $ (A^2-B^2)= (A-B)(A+B) $
2. Square of the sum of two numbers:
  $ (A+B)^2=A^2+B^2+2AB $
3. Square of the difference of two numbers:
  $ (A-B)^2=A^2+B^2-2AB $
We will use the first identity in numerator and 2 &3 identity in the denominator. Then we will cancel the term in both numerator and denominator and simplify the expression.

Complete step by step answer:
Step1:
We are given an expression $ \dfrac{{{x^4} - {y^4}}}{{\left( {{x^4} + 2{x^2}{y^2} + {y^4}} \right)\left( {{x^2} - 2xy + {y^2}} \right)}} $ first we will solve the numerator by using the identity: $(A^2-B^2)= (A-B)(A+B)$
Here $ {A^2} = {x^4};{B^2} = {y^4} $ on substituting the value in identity we will get:
  $ \Rightarrow \dfrac{{\left( {{x^2} - {y^2}} \right)\left( {{x^2} + {y^2}} \right)}}{{\left( {{x^4} + 2{x^2}{y^2} + {y^4}} \right)\left( {{x^2} - 2xy + {y^2}} \right)}} $

Step2:
Now we will solve the denominator by using the identity
$(A+B)^2=A^2+B^2+2AB$
$(A-B)^2=A^2+B^2-2AB$
On using these we will get:
  $ \Rightarrow \dfrac{{\left( {{x^2} - {y^2}} \right)\left( {{x^2} + {y^2}} \right)}}{{{{\left( {{x^2} + {y^2}} \right)}^2}{{\left( {x - y} \right)}^2}}} $
On cancelling the common term $ \left( {{x^2} + {y^2}} \right) $ from the numerator and denominator we will get:
  $ \Rightarrow \dfrac{{\left( {{x^2} - {y^2}} \right)}}{{\left( {{x^2} + {y^2}} \right){{\left( {x - y} \right)}^2}}} $

Step3:
Now we will again apply the identity $(A^2-B^2)= (A-B)(A+B)$ in the numerator we will get:
  $ \Rightarrow \dfrac{{\left( {x - y} \right)\left( {x + y} \right)}}{{\left( {{x^2} + {y^2}} \right){{\left( {x - y} \right)}^2}}} $
On cancelling the term $ \left( {x + y} \right) $ in the numerator and denominator we will get:
  $ \Rightarrow \dfrac{{\left( {x + y} \right)}}{{\left( {{x^2} + {y^2}} \right)}} $
Hence the simplified form is $ \dfrac{{\left( {x + y} \right)}}{{\left( {{x^2} + {y^2}} \right)}} $

Note: In such types of questions by seeing the question students did not get any approach how to solve it. But if we keep in mind the concept of finding identities then we can easily solve such questions in 2-3 minutes. Because there is no calculation in these questions, only the application of identities correctly is required. And factoring out the common term in many cases is also important. In these questions, trying to factor out common terms before applying the identity by doing these things can be easily solved.