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 expression ${\left(-3\right)}^4$ without using a calculator.

Answer to Problem 1ADT

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

Explanation of Solution

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

Obtain the value as follows.

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

Therefore, the value of the expression ${\left(-3\right)}^4$ is $\underset{\_}{81}$.

To determine

To evaluate: The expression $-3^4$ without using a calculator.

Answer to Problem 1ADT

The value of the expression $-3^4$ is $\underset{\_}{-81}$.

Explanation of Solution

Given expression is $-3^4$.

Obtain the value as follows.

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

Therefore, the value of the expression $-3^4$ is $\underset{\_}{-81}$.

To determine

To evaluate: The expression $3^{-4}$ without using a calculator.

Answer to Problem 1ADT

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

Explanation of Solution

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

Obtain the value as follows.

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

Therefore, the value of the expression $3^{-4}$ is $\underset{\_}{\frac{1}{81}}$.

To determine

To evaluate: The expression $\frac{5^{23}}{5^{21}}$ without using a calculator.

Answer to Problem 1ADT

The value of the expression $\frac{5^{23}}{5^{21}}$ is $\underset{\_}{25}$.

Explanation of Solution

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

Obtain the value as follows.

$\begin{array}{c}\frac{5^{23}}{5^{21}}=5^{23-21}\left(\because \frac{a^m}{a^n}=a^{m-n}\right)\\ =5^2\\ =5\times 5\\ =25\end{array}$

Therefore, the value of the expression $\frac{5^{23}}{5^{21}}$ is $\underset{\_}{25}$.

To determine

To evaluate: The expression ${\left(\frac{2}{3}\right)}^{-2}$ without using a calculator.

Answer to Problem 1ADT

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

Explanation of Solution

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

Obtain the value as follows.

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

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

To determine

To evaluate: The expression ${16}^{-\frac{3}{4}}$ without using a calculator.

Answer to Problem 1ADT

The value of the expression ${16}^{-\frac{3}{4}}$ is $\underset{\_}{\frac{1}{8}}$.

Explanation of Solution

Given expression is ${16}^{-\frac{3}{4}}$.

Obtain the value as follows.

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

Therefore, the value of the expression ${16}^{-\frac{3}{4}}$ is $\underset{\_}{\frac{1}{8}}$.