Chapter 7.9, Problem 31WE

a.

To determine

To calculate: The simplified form of the expression ${\left(2^{-1}+2^{-2}\right)}^{-1}$ .

Answer to Problem 31WE

The simplified form of the expression ${\left(2^{-1}+2^{-2}\right)}^{-1}$ is $\frac{1}{400}$ .

Explanation of Solution

Given information:

The expression ${\left(2^{-1}+2^{-2}\right)}^{-1}$ .

Formula used:

The expression $a^{-n}$ is expressed as $\frac{1}{a^n}$ where a is a nonzero real number and n is a positive integer.

Law of exponent for power of quotient is ${\left(\frac{a}{b}\right)}^n=\frac{a^n}{b^n}$ where a andb are nonzero integers and n is a positive integer.

Calculation:

Consider the expression ${\left(2^{-1}+2^{-2}\right)}^{-1}$ .

Recall that expression $a^{-n}$ is expressed as $\frac{1}{a^n}$ where a is a nonzero real number and n is a positive integer.

Apply it,

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

Recall that law of exponent for power of quotient is ${\left(\frac{a}{b}\right)}^n=\frac{a^n}{b^n}$ where a andb are nonzero integers and n is a positive integer.

Apply it,

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

Recall that expression $a^{-n}$ is expressed as $\frac{1}{a^n}$ where a is a nonzero real number and n is a positive integer.

Apply it,

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

Thus, the simplified form of the expression ${\left(2^{-1}+2^{-2}\right)}^{-1}$ is $\frac{4}{3}$ .

b.

To determine

To calculate: The simplified form of the expression ${\left(2^{-1}\cdot 2^{-2}\right)}^{-1}$ .

Answer to Problem 31WE

The simplified form of the expression ${\left(2^{-1}\cdot 2^{-2}\right)}^{-1}$ is $8$ .

Explanation of Solution

Given information:

The expression ${\left(2^{-1}\cdot 2^{-2}\right)}^{-1}$ .

Formula used:

Law of exponent for product of power is $a^m\cdot a^n=a^{m+n}$ where a andb are nonzero integers and n and m are positive integers.

Calculation:

Consider the expression ${\left(2^{-1}\cdot 2^{-2}\right)}^{-1}$ .

Recall that law of exponent for product of power is $a^m\cdot a^n=a^{m+n}$ where a andb are nonzero integers and n and m are positive integers.

Apply it,

  $\begin{array}{c}{\left(2^{-1}\cdot 2^{-2}\right)}^{-1}={\left(2^{-1+\left(-2\right)}\right)}^{-1}\\ ={\left(2^{-3}\right)}^{-1}\end{array}$

Recall that law of exponent for power of power is ${\left(a^m\right)}^n=a^{mn}$ where a is a nonzero integers and n and m are positive integers.

Apply it,

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

Thus, the simplified form of the expression ${\left(2^{-1}\cdot 2^{-2}\right)}^{-1}$ is $8$ .