Chapter T, Problem 1ADT

Evaluate each expression without using a calculator.

1. (a) (−3)⁴
2. (b) −3⁴
3. (c) 3⁻⁴
4. (d) $\frac{5^{23}}{5^{21}}$
5. (e) ${\left(\frac{2}{3}\right)}^{-2}$
6. (f) 16^−3/4

To determine

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

Answer to Problem 1ADT

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

Explanation of Solution

Consider the given expression ${\left(-3\right)}^4$.

Since the power is 4, multiply −3 four times to compute the value of ${\left(-3\right)}^4$.

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

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

To determine

To evaluate: The value of $-3^4$.

Answer to Problem 1ADT

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

Explanation of Solution

Consider the given expression $-3^4$.

Since the power is 4, multiply 3 four times to compute the value of $-3^4$.

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

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

To determine

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

Answer to Problem 1ADT

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

Explanation of Solution

Consider the given expression $3^{-4}$.

Since the power is $-4$ , multiply $\frac{1}{3}$ four times to compute the value of $3^{-4}$.

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

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

To determine

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

Answer to Problem 1ADT

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

Explanation of Solution

Formula used:

The a is real number and m, n are natural numbers then $\frac{a^m}{a^n}=a^{m-n}$.

Calculation:

Consider the given expression $\frac{5^{23}}{5^{21}}$.

Use the above mentioned formula compute the value of $\frac{5^{23}}{5^{21}}$.

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

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

To determine

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

Answer to Problem 1ADT

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

Explanation of Solution

Consider the given expression ${\left(\frac{2}{3}\right)}^{-2}$.

Since the power is $-2$ , multiply $\frac{3}{2}$ two times compute the value of ${\left(\frac{2}{3}\right)}^{-2}$.

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

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

To determine

To evaluate: The value of ${\left(16\right)}^{-\frac{3}{4}}$.

Answer to Problem 1ADT

The value of ${\left(16\right)}^{-\frac{3}{4}}$ is $\frac{1}{8}$.

Explanation of Solution

Consider the given expression ${\left(16\right)}^{-\frac{3}{4}}$.

Since the power is $-\frac{3}{4}$ , multiply $\frac{1}{16}$ three fourth times compute the value of ${\left(16\right)}^{-\frac{3}{4}}$.

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

Thus, the value of ${\left(16\right)}^{-\frac{3}{4}}$ is $\frac{1}{8}$.