Chapter 40, Problem 30A

To determine

(a)

Compute the given signed number ${\left(-2\right)}^5$

Answer to Problem 30A

Value of the given number is, -32

Explanation of Solution

Given:

  ${\left(-2\right)}^5$ is given as a signed number

Calculation:

Here digit (-2) is indicated as power of 5 i.e. fifth power of (-2)

  $\begin{array}{l}{\left(-2\right)}^5\\ =\left(-2\right)\times \left(-2\right)\times \left(-2\right)\times \left(-2\right)\times \left(-2\right)\end{array}$

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 an odd number. Therefore the product will be a negative number.

  $\begin{array}{l}=\left(-2\right)\times \left(-2\right)\times \left(-2\right)\times \left(-2\right)\times \left(-2\right)\\ =(-2\times -2\times -2\times -2\times -2)\\ =-32\end{array}$

Hence the value of the given number is, -32

To determine

(b)

Compute the given signed number ${\left(-6\right)}^2$

Answer to Problem 30A

Value of the given number is, 36

Explanation of Solution

Given:

  ${\left(-6\right)}^2$ is given as a signed number

Calculation:

Here digit (-6) is indicated as power of 2 i.e. square of (-6)

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

Now count the number of negative signs in the given number, if the sum is a even number then the product is positive and if the sum is a odd number then the product is negative.

In this given number, the sum of negative signs is 2 which is a even number. Therefore the product will be a positive number.

  $\begin{array}{l}=\left(-6\right)\times \left(-6\right)\\ =(6\times 6)\\ =36\end{array}$

Hence the value of the given number is, 36

To determine

(c)

Compute the given signed number ${\left(-5\right)}^3$

Answer to Problem 30A

Value of the given number is, -125

Explanation of Solution

Given:

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

Calculation:

Here digit (-5) is indicated as power of 3 i.e. cubic power of (-5)

  $\begin{array}{l}$ {(-5)}^3$=\left(-5\right)\times \left(-5\right)\times \left(-5\right)\end{array}$

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.

  $\begin{array}{l}=\left(-5\right)\times \left(-5\right)\times \left(-5\right)\\ =(-5\times -5\times -5)\\ =-125\end{array}$

Hence the value of the given number is, -125

To determine

(d)

Compute the given signed number ${\left(-2\right)}^6$

Answer to Problem 30A

Value of the given number is, 64

Explanation of Solution

Given:

  ${\left(-2\right)}^6$ is given as a signed number

Calculation:

Here digit (-2) is indicated as power of 6 i.e. sixth power of (-2)

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

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 6 which is a even number. Therefore the product will be a positive number.

  $\begin{array}{l}=\left(-2\right)\times \left(-2\right)\times \left(-2\right)\times \left(-2\right)\times \left(-2\right)\times \left(-2\right)\\ =(-2\times -2\times -2\times -2\times -2\times -2)\\ =64\end{array}$

Hence the value of the given number is, 64

To determine

(e)

Compute the given signed number ${\left(-1.6\right)}^2$

Answer to Problem 30A

Value of the given number is, 2.56

Explanation of Solution

Given:

  ${\left(-1.6\right)}^2$ is given as a signed number

Calculation:

Here digit (-1.6) is indicated as power of 2 i.e. square of (-1.6)

  $\begin{array}{l}{\left(-1.6\right)}^2\\ =\left(-1.6\right)\times \left(-1.6\right)\end{array}$

Now count the number of negative signs in the given number, if the sum is a even number then the product is positive and if the sum is a odd number then the product is negative.

In this given number, the sum of negative signs is 2 which is a even number. Therefore the product will be a positive number.

  $\begin{array}{l}=\left(-1.6\right)\times \left(-1.6\right)\\ =(1.6\times 1.6)\\ =2.56\end{array}$

Hence the value of the given number is, 2.56