Chapter 4.3, Problem 23WE

To determine

To multiply given expression assuming variable exponents represent positive integers

Answer to Problem 23WE

  $xy{\left(x-y\right)}^2=x^{3}y-2x^2{y}^2+x{y}^3$

Explanation of Solution

Given information : 

Given expression $xy{\left(x-y\right)}^2$

Concept used:

Using identity

  ${\left(a-b\right)}^2=a^2-2\left(a\times b\right)+{\left(b\right)}^2$

And

Basic rule of multiplication of monomial and binomial

  $x\left(a+b\right)=\left(x\times a\right)+\left(x\times b\right)$

Rule of exponents

  $a^m.a^n=a^{m+n}$

Explanation: 

  $\begin{array}{l}xy{\left(x-y\right)}^2\\ =xy\left({\left(x\right)}^2-2xy+{\left(y\right)}^2\right)\\ =xy\left(x^2-2xy+y^2\right)\\ =\left(xy\times x^2\right)-\left(xy\times 2xy\right)+\left(xy\times y^2\right)\\ =x^{1+2}y-2x^{1+1}{y}^{1+1}+x{y}^{1+2}\\ =x^{3}y-2x^2{y}^2+x{y}^3\end{array}$