Chapter 4.4, Problem 24WE

(a).

To determine

To find: the GCF of the given monomials.

Answer to Problem 24WE

The GCF of $22x{y}^2{z}^2,\text{ 33}{x}^{2}y{z}^2,$ and $\text{44}{x}^{2}yz$ is $11xyz$ .

Explanation of Solution

Given information:

The given monomials:

  $22x{y}^2{z}^2,\text{ 33}{x}^{2}y{z}^2,\text{ 44}{x}^{2}yz$

Calculation:

The prime factorization of the coefficients are:

  $\begin{array}{l}22=2\cdot 11\\ 33=3\cdot 11\\ 44=2^2\cdot 11\end{array}$

The GCF of $22,\text{ 33}$ and $44$ is $11$

Compare the powers of each variable occurring in all three monomials. Use the power with the least exponent.

Compare $x$ and $x^2$ . Use $x$ .

Compare $y^2$ and $y$ . Use $y$ .

Compare $z^2$ and $z$ . Use $z$ .

  $\text{GCF}=11\cdot x\cdot y\cdot z=11xyz$

Therefore, the GCF of $22x{y}^2{z}^2,\text{ 33}{x}^{2}y{z}^2,$ and $\text{44}{x}^{2}yz$ is $11xyz$ .

(b).

To determine

The LCM of the given monomials.

Answer to Problem 24WE

The LCM of $22x{y}^2{z}^2,\text{ 33}{x}^{2}y{z}^2,$ and $\text{44}{x}^{2}yz$ is $132x^2{y}^2{z}^2$ .

Explanation of Solution

Given information:

The given monomials:

  $22x{y}^2{z}^2,\text{ 33}{x}^{2}y{z}^2,\text{ 44}{x}^{2}yz$

Calculation:

As per problem,

The prime factorization of the coefficients are:

  $\begin{array}{l}22=2\cdot 11\\ 33=3\cdot 11\\ 44=2^2\cdot 11\end{array}$

The LCM of $22,\text{ 33}$ and $44$ is

  $2^2\cdot 3\cdot 11=132$

Compare the powers of each variable occurring in either monomial. Use the power with the greatest exponent.

Compare $x$ and $x^2$ . Use $x^2$ .

Compare $y^2$ and $y$ . Use $y^2$ .

Compare $z^2$ and $z$ . Use $z^2$ .

  $\text{LCM}=132\cdot x^2\cdot y^2\cdot z^2=132x^2{y}^2{z}^2$

Therefore, the LCM of $22x{y}^2{z}^2,\text{ 33}{x}^{2}y{z}^2,$ and $\text{44}{x}^{2}yz$ is $132x^2{y}^2{z}^2$ .