Chapter 7.3, Problem 3P

a.

To determine

To find: The simplified form of the expression ${\left(7m^9\right)}^3$ .

Answer to Problem 3P

The required simplified form is $\frac{1}{x^{22}}$ .

Explanation of Solution

Given information:

The given expression ${\left(7m^9\right)}^3$ .

Calculation:

  $\begin{array}{l}={\left(7m^9\right)}^3\\ =7^3{\left(m^9\right)}^3\end{array}$

(Raise each factor to the second power)

  $=7^3{m}^{27}$

(Multiply the exponents of a power raised to a power)

  $=343m^{27}$

(Simplify)

b.

To determine

To find: The simplified form of the expression ${\left(2z\right)}^{-4}$ .

Answer to Problem 3P

The required simplified form is $\frac{1}{16z^4}$ .

Explanation of Solution

Given information:

The given expression ${\left(2z\right)}^{-4}$ .

Calculation:

  $\begin{array}{l}={\left(2z\right)}^{-4}\\ =2^{-4}{z}^{-4}\end{array}$

(Raise each factor to the second power)

  $\begin{array}{l}=\frac{1}{16}\cdot z^{-4}\\ =\frac{1}{16}\cdot \frac{1}{z^4}\end{array}$

(The expression using only positive exponents)

  $=\frac{1}{16z^4}$

(Simplify)

c.

To determine

To find: The simplified form of the expression ${\left(3g^4\right)}^{-2}$ .

Answer to Problem 3P

The required simplified form is $\frac{1}{9g^8}$ .

Explanation of Solution

Given information:

The given expression ${\left(3g^4\right)}^{-2}$ .

Calculation:

  $\begin{array}{l}={\left(3g^4\right)}^{-2}\\ =3^{-2}{g}^{4\cdot \left(-2\right)}\end{array}$

(Raise each factor to the second power)

  $=3^{-2}{g}^{-8}$

(Multiply the exponents of a power raised to a power)

  $=\frac{1}{9}\cdot \frac{1}{g^8}$

(The expression using only positive exponents)

  $=\frac{1}{9g^8}$

(Simplify)