Chapter 1.2, Problem 45E

(a)

To determine

To simplify: given expression $\frac{6y^{3}z}{2y{z}^2}$ .

Answer to Problem 45E

  $\frac{6y^{3}z}{2y{z}^2}=\frac{3y^2}{z}$

Explanation of Solution

Given information:

Given expression

  $\frac{6y^{3}z}{2y{z}^2}$

Calculation:

Consider the given expression.

  $\frac{6y^{3}z}{2y{z}^2}$

Now, use the second law of exponents, which states

  $\frac{a^m}{a^n}=a^{m-n}$

And combine like terms

  $\begin{array}{c}\frac{6y^{3}z}{2y{z}^2}=\frac{6}{2}{y}^{3-1}{z}^{1-2}\\ =3y^2{z}^{-1}\\ =\frac{3y^2}{z}\end{array}$

Hence,

  $\frac{6y^{3}z}{2y{z}^2}=\frac{3y^2}{z}$

(b)

To determine

To evaluate: the given expression $\frac{{\left(x{y}^2{z}^3\right)}^4}{{\left(x^2{y}^{2}z\right)}^3}$ .

Answer to Problem 45E

  $\frac{{\left(x{y}^2{z}^3\right)}^4}{{\left(x^2{y}^{2}z\right)}^3}=\frac{y^2{z}^9}{x^2}$

Explanation of Solution

Given information:

Given expression

  $\frac{{\left(x{y}^2{z}^3\right)}^4}{{\left(x^2{y}^{2}z\right)}^3}$

Calculation:

Consider the given expression.

  $\frac{{\left(x{y}^2{z}^3\right)}^4}{{\left(x^2{y}^{2}z\right)}^3}$

Now. Simplify using law of exponents ${\left(ab\right)}^n=a^n{b}^n$ and ${\left(a^m\right)}^n=a^{mn}$ .

  $\begin{array}{c}\frac{{\left(x{y}^2{z}^3\right)}^4}{{\left(x^2{y}^{2}z\right)}^3}=\frac{\left\{x^4{\left(y^2\right)}^4{\left(z^3\right)}^4\right\}}{\left\{{\left(x^2\right)}^3{\left(y^2\right)}^3{\left(z\right)}^3\right\}}\\ =\frac{x^4{y}^8{z}^{12}}{x^6{y}^6{z}^3}\end{array}$

Now, use the second law of exponents, which states

  $\frac{a^m}{a^n}=a^{m-n}$

And combine like terms

  $\begin{array}{c}\frac{x^4{y}^8{z}^{12}}{x^6{y}^6{z}^3}=x^{4-6}{y}^{8-6}{z}^{12-3}\\ =x^{-2}{y}^2{z}^9\\ =\frac{y^2{z}^9}{x^2}\end{array}$

Hence,

  $\frac{{\left(x{y}^2{z}^3\right)}^4}{{\left(x^2{y}^{2}z\right)}^3}=\frac{y^2{z}^9}{x^2}$