Chapter 1.1, Problem 66E

(a)

To determine

To evaluate: the expression $\left|\sqrt{5}-5\right|$ .

Answer to Problem 66E

  $\left|\sqrt{5}-5\right|=2.7640$

Explanation of Solution

Given information:

An expression is given as $\left|\sqrt{5}-5\right|$ .

Formula used:

Absolute value property:

For any real number a,

  $\begin{array}{l}\left|a\right|=a\ge 0\\ \left|-a\right|=\left|a\right|=a\end{array}$

Calculation:

Consider the given expression.

  $\left|\sqrt{5}-5\right|$

Value of $\sqrt{5}$ is 2.2360.

Substitute the value in $\left|\sqrt{5}-5\right|$ .

  $\begin{array}{c}\left|\sqrt{5}-5\right|=\left|2.2360-5\right|\\ =\left|-2.7640\right|\end{array}$

Now, compare the expression with $\left|-a\right|$ :

  $\begin{array}{c}\left|-a\right|=\left|-2.7640\right|\\ a=2.7640\end{array}$

Now, by absolute value property:

  $\left|-a\right|=\left|a\right|=a$

Therefore, $\left|-2.7640\right|=2.7640$

Hence, $\left|\sqrt{5}-5\right|=2.7640$

(b)

To determine

To evaluate: the expression $\left|10-\pi \right|$ .

Answer to Problem 66E

  $\left|10-\pi \right|=6.8571$

Explanation of Solution

Given information:

An expression is given as $\left|10-\pi \right|$ .

Formula used:

Absolute value property:

For any real number a,

  $\begin{array}{l}\left|a\right|=a\ge 0\\ \left|-a\right|=\left|a\right|=a\end{array}$

Calculation:

Consider the given expression.

Value of $\pi$ is 3.1428.

Substitute the value in $\left|10-\pi \right|$ .

  $\begin{array}{c}\left|10-\pi \right|=\left|10-3.1428\right|\\ =\left|6.8571\right|\end{array}$

Now, compare the expression with $\left|a\right|$ :

  $\begin{array}{l}\left|a\right|=\left|6.8571\right|\\ a=6.8571\end{array}$

Now, by absolute value property:

  $\left|a\right|=a\ge 0$

Therefore, $\left|6.8571\right|=6.8571$

Hence, $\left|10-\pi \right|=6.8571$ .