Question

The product of \[\left( {{x^2} + {y^2}} \right)\left( {{x^2} - {y^2}} \right)\] is ______________.
A) \[{x^4} - {y^4}\]
B) \[{x^4} + {y^4}\]
C) \[{x^4} + 2xy - {y^4}\]
D) None of these

Answer 1

Hint: Here we have to know the formula \[{a^2} - {b^2} = \left( {a + b} \right)\left( {a - b} \right)\] where \[a\] and \[b\] are any non-zero real numbers. Using this formula, the given product can be rewritten in simple difference. This property is useful when we want to convert the product like in RHS of this formula as a difference of squares like in LHS of this equation.

Complete step-by-step solution:
Algebra is based on the concept of unknown values called variables, but arithmetic is based entirely on known number values. The basic laws of Algebra are the associative, commutative and distributive laws. They help explain the relationship between number operations and lend towards simplifying equations or solving them.

Now let us solve the given problem
Given \[\left( {{x^2} + {y^2}} \right)\left( {{x^2} - {y^2}} \right)\]----(1)
Since we know that \[{a^2} - {b^2} = \left( {a + b} \right)\left( {a - b} \right)\]. In the expression (1), \[a = {x^2}\] and \[b = {y^2}\] then the expression (1) can be written as
\[\left( {{x^2} + {y^2}} \right)\left( {{x^2} - {y^2}} \right) = {\left( {{x^2}} \right)^2} - {\left( {{y^2}} \right)^2}\]
Simplifying the above equation, we get
\[\left( {{x^2} + {y^2}} \right)\left( {{x^2} - {y^2}} \right) = {x^4} - {y^4}\]

Hence the Option A is the correct answer.

Note: Some important algebra formulas that have to remembered to solve the problems in simple way:
- \[{\left( {a + b} \right)^2} = {a^2} + {b^2} + 2ab\]
- \[{\left( {a - b} \right)^2} = {a^2} + {b^2} - 2ab\]
- \[{a^2} - {b^2} = \left( {a + b} \right)\left( {a - b} \right)\]
- \[{\left( {a - b} \right)^3} = {a^3} - {b^3} - 3ab\left( {a - b} \right)\]
- \[{\left( {a + b} \right)^3} = {a^3} + {b^3} + 3ab\left( {a + b} \right)\]
- \[{a^3} + {b^3} = \left( {a + b} \right)\left( {{a^2} + {b^2} - ab} \right)\]
- \[{a^3} - {b^3} = \left( {a - b} \right)\left( {{a^2} + {b^2} + ab} \right)\]