Chapter 1.2, Problem 17E

(a)

To determine

To evaluate: given expression ${\left(\frac{5}{3}\right)}^0{2}^{-1}$ .

Answer to Problem 17E

  ${\left(\frac{5}{3}\right)}^0{2}^{-1}=\frac{1}{2}$

Explanation of Solution

Given information:

Given expression

  ${\left(\frac{5}{3}\right)}^0{2}^{-1}$

Calculation:

Consider the given expression.

  ${\left(\frac{5}{3}\right)}^0{2}^{-1}$

Use the law of the exponents to simplify the expression.

  ${\left(\frac{5}{3}\right)}^0\cdot 2^{-1}=1\cdot 2^{-1}$

Use zero-exponent rule $a^0=1$

  $=2^{-1}$

Use negative exponent rule $a^{-n}=\frac{1}{a^n}$

  $=\frac{1}{2}$

(b)

To determine

To evaluate: the given expression $\frac{2^{-3}}{3^0}$ .

Answer to Problem 17E

  $\frac{2^{-3}}{3^0}=\frac{1}{8}$

Explanation of Solution

Given information:

Given expression

  $\frac{2^{-3}}{3^0}$ .

Calculation:

Consider the given expression.

  $\frac{2^{-3}}{3^0}$

Use the law of the exponents to simplify the expression.

  $\frac{2^{-3}}{3^0}=\frac{2^{-3}}{1}$

Use zero-exponent rule $a^0=1$

  $2^{-3}$

Use negative exponent rule $a^{-n}=\frac{1}{a^n}$ .

  $\frac{1}{2^3}=\frac{1}{8}$

(c)

To determine

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

Answer to Problem 17E

  ${\left(\frac{1}{4}\right)}^{-2}=16$

Explanation of Solution

Given information:

Given expression

  ${\left(\frac{1}{4}\right)}^{-2}$

Calculation:

Consider the given expression.

  $\frac{3}{3^{-2}}$

Use the law of the exponents to simplify the expression.

Use negative exponent rule $a^{-n}=\frac{1}{a^n}$

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

Reciprocal of reciprocal of number is itself

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