Chapter 1.2, Problem 15E

(a)

To determine

To evaluate: given expression $-3^2$ .

Answer to Problem 15E

  $-9$

Explanation of Solution

Given information:

Given expression

  $-3^2$

Calculation:

We want to evaluate

  $-3^2$

Because there are no parentheses, evaluate the exponent first, and then multiply by $-1$ . That is, we should first rewrite our expression as

  $-3^2=-\left(3^2\right)$

By the definition of exponential notation, which states that

  $a^n=\underset{n\text{factors}}{\underbrace{a\cdot a\cdot a\cdot \cdot \cdot \cdot \cdot a}}{a}^n=\underset{n\text{factors}}{\underbrace{a\cdot a\cdot a\cdot \cdot \cdot \cdot \cdot a}},$

We can say that $3^2$ is simply 3 multiplied by itself 2 times.

  $\begin{array}{c}{3}^2=3\cdot 3\\ 3^2=9\end{array}$

Now, put this back into our original expression and we get,

  $\begin{array}{l}-3^2=-\left(3^2\right)\\ -3^2=-\left(9\right)\\ -3^2=-9\end{array}$

(b)

To determine

To evaluate: the given expression ${\left(-3\right)}^2$ .

Answer to Problem 15E

  ${\left(-3\right)}^2=9$

Explanation of Solution

Given information:

Given expression

  ${\left(-3\right)}^2$ .

Calculation:

Because of the parentheses, the power is applied to $-3$ . By the definition of exponential notation, which states that,

  $a^n=\underset{n\text{factors}}{\underbrace{a\cdot a\cdot a\cdot \cdot \cdot \cdot \cdot a}}{a}^n=\underset{n\text{factors}}{\underbrace{a\cdot a\cdot a\cdot \cdot \cdot \cdot \cdot a}},$

The expression simply represents $-3$ multiplied by itself 2 times.

  $\begin{array}{l}{\left(-3\right)}^2=\left(-3\right)\cdot \left(-3\right)\\ {\left(-3\right)}^2=9\end{array}$

(c)

To determine

To evaluate: given expression ${\left(\frac{1}{3}\right)}^4{\left(-3\right)}^2$ .

Answer to Problem 15E

  ${\left(\frac{1}{3}\right)}^4{\left(-3\right)}^2=\left(\frac{1}{9}\right)$ .

Explanation of Solution

Given information:

Given expression

  ${\left(\frac{1}{3}\right)}^4{\left(-3\right)}^2$

Calculation:

Consider the expression

  ${\left(\frac{1}{3}\right)}^4{\left(-3\right)}^2$

Here, we will again use the definition of exponential notation. Recall that this states

  $a^n=\underset{n\text{factors}}{\underbrace{a\cdot a\cdot a\cdot \cdot \cdot \cdot \cdot a}}{a}^n=\underset{n\text{factors}}{\underbrace{a\cdot a\cdot a\cdot \cdot \cdot \cdot \cdot a}},$

Expand the expression into the multiplication it represents.

  $\begin{array}{c}{\left(\frac{1}{3}\right)}^4{\left(-3\right)}^2=\left(\frac{1}{3}\right)\cdot \left(\frac{1}{3}\right)\cdot \left(\frac{1}{3}\right)\cdot \left(\frac{1}{3}\right)\cdot \left(-3\right)\cdot \left(-3\right)\\ =\left(\frac{1}{81}\right)\cdot \left(9\right)\\ =\left(\frac{1}{9}\right)\end{array}$