Chapter 6.1, Problem 14WE

(a)

To determine

To simplify: The given expression and identify if the expression represents the real number.

Answer to Problem 14WE

The simplified value of the given expression is $ai$ . The number $ai$ is not the real number.

Explanation of Solution

Given:

The given expression is

  $\sqrt{-a^2}$

Calculation:

The given expression can be simplified as follows.

  $\begin{array}{l}=\sqrt{-a^2}\\ =\sqrt{-a\times a}\\ =a\sqrt{-1}\\ =ai\end{array}$

Any number measurable on the number line whether positive, negative, and decimal or in fractions is called the real number.

Here, the number $ai$ may or may not be the real number.

Therefore, the simplified value of the given expression is $ai$ . The number $ai$ is not the real number.

(b)

To determine

To simplify: The given expression and identify if the expression represents the real number.

Answer to Problem 14WE

The simplified value of the given expression is $a$ . The number $a$ will be the real number.

Explanation of Solution

Given:

The given expression is

  $\sqrt{{\left(-a\right)}^2}$

Calculation:

The given expression can be simplified as follows.

  $\begin{array}{l}=\sqrt{{\left(-a\right)}^2}\\ =\sqrt{\left(-a\right)\times \left(-a\right)}\\ =a\end{array}$

Any number measurable on the number line whether positive, negative, and decimal or in fractions is called the real number.

Here, the number $a$ will be the real number because the square of any number can not be negative.

Therefore, the simplified value of the given expression is $a$ . The number $a$ will be the real number.

(c)

To determine

To simplify: The given expression and identify if the expression represents the real number.

Answer to Problem 14WE

The simplified value of the given expression is $a$ . The number $a$ will be the real number.

Explanation of Solution

Given:

The given expression is

  $\sqrt[4]{a^4}$

Calculation:

The given expression can be simplified as follows.

  $\begin{array}{l}=\sqrt[4]{a^4}\\ =\sqrt[4]{a\times a\times a\times a}\\ =a\end{array}$

Any number measurable on the number line whether positive, negative, and decimal or in fractions is called the real number.

Here, the number $a$ will be the real number because it does not consist of negative sign.

Therefore, the simplified value of the given expression is $a$ . The number $a$ will be the real number.

(d)

To determine

To simplify: The given expression and identify if the expression represents the real number.

Answer to Problem 14WE

The simplified value of the given expression is $a^3$ . The number $a^3$ will be the real number.

Explanation of Solution

Given:

The given expression is

  $\sqrt{a^6}$

Calculation:

The given expression can be simplified as follows.

  $\begin{array}{l}=\sqrt{a^6}\\ =a^{6\times \frac{1}{2}}\\ =a^3\end{array}$

Any number measurable on the number line whether positive, negative, and decimal or in fractions is called the real number.

Here, the number $a^3$ will be the real number because $a$ is positive here.

Therefore, the simplified value of the given expression is $a^3$ . The number $a^3$ will be the real number.