Chapter 4.3, Problem 36WE

To determine

To calculate: The multiplication of the term ${\left(x^{2n}-y^n\right)}^2$ .

Answer to Problem 36WE

The required multiplication is $x^{4n}-2x^{2n}{y}^n+y^{2n}$ .

Explanation of Solution

Given information:

The algebraic term is ${\left(x^{2n}-y^n\right)}^2$ .

Formula Used:

Distribution property is

  $\begin{array}{c}\left(a+b\right)\left(c+d\right)=a\left(c+d\right)+b\left(c+d\right)\\ =ac+ad+bc+bd\end{array}$

Calculation:

Consider the given algebraic term ${\left(x^{2n}-y^n\right)}^2$ .

Now multiply. Use the distribution property.

  $\begin{array}{c}{\left(x^{2n}-y^n\right)}^2=\left(x^{2n}-y^n\right)\left(x^{2n}-y^n\right)\\ =x^{2n}\left(x^{2n}-y^n\right)-y^n\left(x^{2n}-y^n\right)\\ =x^{4n}-x^{2n}{y}^n-x^{2n}{y}^n+y^{2n}\\ =x^{4n}-2x^{2n}{y}^n+y^{2n}\end{array}$

Hence, the required multiplication is $x^{4n}-2x^{2n}{y}^n+y^{2n}$ .