Chapter 1, Problem 2T

(a)

To determine

To evaluate: The expression ${\left(-3\right)}^4$.

Answer to Problem 2T

The value of ${\left(-3\right)}^4$ is 81.

Explanation of Solution

Given:

The expression is ${\left(-3\right)}^4$.

Calculation:

In the given expression $-3$ is a negative integer and 4 is the exponent of base $-3$ so the exponential notation of this expression is,

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

Thus, the value of ${\left(-3\right)}^4$ is 81.

(b)

To determine

To evaluate: The expression $-3^4$.

Answer to Problem 2T

The value of $-3^4$ is $-81$.

Explanation of Solution

Given:

The expression is $-3^4$.

Calculation:

In the given expression 3 is the positive integer and 4 is the exponent of base 3 so the exponential notation is,

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

Thus, the value of $-3^4$ is $-81$.

(c)

To determine

To evaluate: The expression $3^{-4}$.

Answer to Problem 2T

The value of $3^{-4}$ is $\frac{1}{81}$.

Explanation of Solution

Given:

The expression is $3^{-4}$.

Calculation:

In the given expression 3 is positive integer and $-4$ is negative exponent of base 3 so by the negative exponents is,

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

Thus, the value of $3^{-4}$ is $\frac{1}{81}$.

(d)

To determine

To evaluate: The expression $\frac{5^{23}}{5^{21}}$.

Answer to Problem 2T

The value of $\frac{5^{23}}{5^{21}}$ is 25.

Explanation of Solution

Given:

The expression is $\frac{5^{23}}{5^{21}}$.

Calculation:

The above expression shows that division of two powers of the same number so subtract the exponents,

$\begin{array}{c}\frac{5^{23}}{5^{21}}=5^{23-21}\\ =5^2\\ =25\end{array}$

Thus, the value of $\frac{5^{23}}{5^{21}}$ is 25.

(e)

To determine

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

Answer to Problem 2T

The value of ${\left(\frac{2}{3}\right)}^{-2}$ is $\frac{9}{4}$.

Explanation of Solution

Given:

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

Calculation:

To raise a fraction to a negative power, invert the fraction and change the sign of the exponent in the above expression,

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

Thus, the value of $\frac{5^{23}}{5^{21}}$ is $\frac{9}{4}$.

(f)

To determine

To evaluate: The expression ${16}^{-3/4}$.

Answer to Problem 2T

The value of ${16}^{-3/4}$ is $\frac{1}{8}$.

Explanation of Solution

Given:

The expression is ${16}^{-3/4}$.

Calculation:

Use exponential law to solve the given expression,

$\begin{array}{c}{16}^{-3/4}={\left({16}^{1/4}\right)}^{-3}\\ ={\left(\sqrt[4]{2\cdot 2\cdot 2\cdot 2}\right)}^{-3}\\ ={\left(2\right)}^{-3}\end{array}$

In the above solution of expression 2 is positive integer and $-3$ negative exponent so by the negative exponents,

$\begin{array}{c}{16}^{-3/4}={\left(2\right)}^{-3}\\ =\frac{1}{2^3}\\ =\frac{1}{2\cdot 2\cdot 2}\\ =\frac{1}{8}\end{array}$

Thus, the value of ${16}^{-3/4}$ is $\frac{1}{8}$.