Chapter 7.3, Problem 2P

a.

To determine

To find: The simplified form of the expression $x^2{\left(x^6\right)}^{-4}$ .

Answer to Problem 2P

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

Explanation of Solution

Given information:

The given expression $x^2{\left(x^6\right)}^{-4}$ .

Calculation:

  $\begin{array}{l}=x^2{\left(x^6\right)}^{-4}\\ =x^2{x}^{6\cdot \left(-4\right)}\end{array}$

(Multiply exponents when raising a power to a power)

  $\begin{array}{l}=x^2{x}^{-24}\\ =x^{2+\left(-24\right)}\end{array}$

(Add exponents when multiplying powers with the same base)

  $\begin{array}{l}=x^{-22}\\ =\frac{1}{x^{22}}\end{array}$

(The expression using only positive exponents)

b.

To determine

To find: The simplified form of the expression $w^{-2}{\left(w^{\frac{5}{3}}\right)}^3$ .

Answer to Problem 2P

The required simplified form is $w^3$ .

Explanation of Solution

Given information:

The given expression $w^{-2}{\left(w^{\frac{5}{3}}\right)}^3$ .

Calculation:

  $\begin{array}{l}=w^{-2}{\left(w^{\frac{5}{3}}\right)}^3\\ =w^{-2}{w}^{\frac{5}{3}\cdot 3}\end{array}$

(Multiply exponents when raising a power to a power)

  $\begin{array}{l}=w^{-2}{w}^5\\ =w^{-2+5}\end{array}$

(Add exponents when multiplying powers with the same base)

  $=w^3$

(Simplify)

c.

To determine

To find: The simplified form of the expression ${\left(s^{-5}\right)}^{-\frac{1}{2}}\left(s^{\frac{3}{2}}\right)$ .

Answer to Problem 2P

The required simplified form is $s^4$ .

Explanation of Solution

Given information:

The given expression ${\left(s^{-5}\right)}^{-\frac{1}{2}}\left(s^{\frac{3}{2}}\right)$ .

Calculation:

  $\begin{array}{l}={\left(s^{-5}\right)}^{-\frac{1}{2}}\left(s^{\frac{3}{2}}\right)\\ =s^{-5\cdot \left(-\frac{1}{2}\right)}{s}^{\frac{3}{2}}\end{array}$

(Multiply exponents when raising a power to a power)

  $\begin{array}{l}=s^{\frac{5}{2}}{s}^{\frac{3}{2}}\\ =s^{\frac{5}{2}+\frac{3}{2}}\end{array}$

(Add exponents when multiplying powers with the same base)

  $=s^4$

(Simplify)