Chapter 40, Problem 32A

To determine

(a)

Compute the given signed number ${\left(-\frac{2}{3}\right)}^3$

Answer to Problem 32A

Value of the given number is, $-\frac{8}{27}$

Explanation of Solution

Given:

  ${\left(-\frac{2}{3}\right)}^3$ is given as a signed number

Calculation:

Here digit $\left(-\frac{2}{3}\right)$ is indicated as power of 3 i.e. cubic power of $\left(-\frac{2}{3}\right)$

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

Now count the number of negative signs in the given number:

1. If the sum is a even number then the product is positive.

2. if the sum is a odd number then the product is negative.

In this given number, the sum of negative signs is 3 which is an odd number. Therefore the product will be a negative number.

  $\begin{array}{l}=\left(-\frac{2}{3}\right)\times \left(-\frac{2}{3}\right)\times \left(-\frac{2}{3}\right)\\ =\left(-\frac{2}{3}\times -\frac{2}{3}\times -\frac{2}{3}\right)\\ =\left[-\left(\frac{2\times 2\times 2}{3\times 3\times 3}\right)\right]\\ =-\frac{8}{27}\end{array}$

Hence the value of the given number is, $-\frac{8}{27}$

To determine

(b)

Compute the given signed number ${\left(-1.038\right)}^{-5}$

Answer to Problem 32A

Value of the given number is, -0.83

Explanation of Solution

Given:

  ${\left(-1.038\right)}^{-5}$ is given as a signed number

Calculation:

Here digit (-1.038) is indicated as power of 5 i.e. fifth power of (-1.038)

  $\begin{array}{l}={\left(-1.038\right)}^{-5}\\ =\frac{1}{{\left(-1.038\right)}^5}\\ =\frac{1}{\left(-1.038\right)\times \left(-1.038\right)\times \left(-1.038\right)\times \left(-1.038\right)\times \left(-1.038\right)}\end{array}$

Now count the number of negative signs in the given number:

1. If the sum is a even number then the product is positive.

2. if the sum is a odd number then the product is negative.

In this given number, the sum of negative signs is 5 which is a odd number. Therefore the product will be a negative number.

  $\begin{array}{l}=\frac{1}{\left(-1.038\right)\times \left(-1.038\right)\times \left(-1.038\right)\times \left(-1.038\right)\times \left(-1.038\right)}\\ =-\frac{1}{1.205}\\ =-0.83\end{array}$

Hence the value of the given number is, -0.83

To determine

(c)

Compute the given signed number ${\left(17.66\right)}^{-2}$

Answer to Problem 32A

Value of the given number is, 0.0032

Explanation of Solution

Given:

  ${\left(17.66\right)}^{-2}$ is given as a signed number

Calculation:

Here digit (17.66) is indicated as power of 2 i.e. square of (17.66)

  $\begin{array}{l}$ {(17.66)}^{-2}$=\frac{1}{(17.66)\times (17.66)}\end{array}$

All the digits in the given problem have positive signs. As there is no negative sign in the given number .So the product will also be a positive sign.

  $\begin{array}{l}=\frac{1}{(17.66)\times (17.66)}\\ =\frac{1}{311.87}\\ =0.0032\end{array}$

Hence the value of the given number is, 0.0032

To determine

(d)

Compute the given signed number ${\left(-0.83\right)}^{-3}$

Answer to Problem 32A

Value of the given number is, -1.748

Explanation of Solution

Given:

  ${\left(-0.83\right)}^{-3}$ is given as a signed number

Calculation:

Here digit (-0.83) is indicated as power of 3 i.e. cubic power of (-0.83)

  $\begin{array}{l}={\left(-0.83\right)}^{-3}\\ =\frac{1}{{\left(-0.83\right)}^3}\\ =\frac{1}{\left(-0.83\right)\times \left(-0.83\right)\times \left(-0.83\right)}\end{array}$

Now count the number of negative signs in the given number:

1. If the sum is a even number then the product is positive.

2. if the sum is a odd number then the product is negative.

In this given number, the sum of negative signs is 3 which is a odd number. Therefore the product will be a negative number.

  $\begin{array}{l}=\frac{1}{\left(-0.83\right)\times \left(-0.83\right)\times \left(-0.83\right)}\\ =-\left(\frac{1}{0.572}\right)\\ =-1.748\end{array}$

Hence the value of the given number is, -1.748

To determine

(e)

Compute the given signed number ${\left(-1.038\right)}^{-5}$

Answer to Problem 32A

Value of the given number is, -0.000728

Explanation of Solution

Given:

  ${\left(-6.087\right)}^{-4}$ is given as a signed number

Calculation:

Here digit (-6.087) is indicated as power of 4 i.e. fourth power of (-6.087)

  $\begin{array}{l}={\left(-6.087\right)}^{-4}\\ =\frac{1}{{\left(-6.087\right)}^4}\\ =\frac{1}{\left(-6.087\right)\times \left(-6.087\right)\times \left(-6.087\right)\times \left(-6.087\right)}\end{array}$

Now count the number of negative signs in the given number:

1. If the sum is a even number then the product is positive.

2. if the sum is a odd number then the product is negative.

In this given number, the sum of negative signs is 4 which is a even number. Therefore the product will be a positive number.

  $\begin{array}{l}=\frac{1}{\left(-6.087\right)\times \left(-6.087\right)\times \left(-6.087\right)\times \left(-6.087\right)}\\ =-\frac{1}{1372.81}\\ =-0.000728\end{array}$

Hence the value of the given number is, -0.000728