Chapter 6.2, Problem 28WE

To determine

To calculate: The simplified form of the expression ${\left(\frac{3z}{y}\right)}^3\cdot \frac{2yz}{15}$.

Answer to Problem 28WE

The simplified form of the expression ${\left(\frac{3z}{y}\right)}^3\cdot \frac{2yz}{15}$ is $\frac{18z^4}{5y^2}$.

Explanation of Solution

Given information:

The expression ${\left(\frac{3z}{y}\right)}^3\cdot \frac{2yz}{15}$.

Formula used:

Law of exponent for a quotient raised to power m where m is a positive integer is ${\left(\frac{a}{b}\right)}^m=\frac{a^m}{b^m}$.

Fractions are multiplied as $\frac{a}{b}\cdot \frac{c}{d}=\frac{ab}{cd}$.

Law of exponent for a quotient raised to power m where m is a positive integer is ${\left(\frac{a}{b}\right)}^m=\frac{a^m}{b^m}$.

Law of exponent for a product of two terms raised to power m where m is a positive integer is ${\left(ab\right)}^m=a^m{b}^m$.

Calculation:

Consider the expression ${\left(\frac{3z}{y}\right)}^3\cdot \frac{2yz}{15}$.

Recall that law of exponent for a quotient raised to power m where m is a positive integer is ${\left(\frac{a}{b}\right)}^m=\frac{a^m}{b^m}$.

Apply it,

  ${\left(\frac{3z}{y}\right)}^3\cdot \frac{2yz}{15}=\frac{{\left(3z\right)}^3}{y^3}\cdot \frac{2yz}{15}$

Recall that law of exponent for a product of two terms raised to power m where m is a positive integer is ${\left(ab\right)}^m=a^m{b}^m$.

  $\begin{array}{c}{\left(\frac{3z}{y}\right)}^3\cdot \frac{2yz}{15}=\frac{{\left(3z\right)}^3}{y^3}\cdot \frac{2yz}{15}\\ =\frac{3^3{z}^3}{y^3}\cdot \frac{2yz}{15}\\ =\frac{27z^3}{y^3}\cdot \frac{2yz}{15}\end{array}$

Recall that fractions are multiplied as $\frac{a}{b}\cdot \frac{c}{d}=\frac{ab}{cd}$.

Apply it,

  $\begin{array}{c}{\left(\frac{3z}{y}\right)}^3\cdot \frac{2yz}{15}=\frac{27z^3}{y^3}\cdot \frac{2yz}{15}\\ =\frac{18z^{4}y}{5y^3}\\ =\frac{18z^4}{5y^2}\end{array}$

Thus, the simplified form of the expression ${\left(\frac{3z}{y}\right)}^3\cdot \frac{2yz}{15}$ is $\frac{18z^4}{5y^2}$.