Chapter 1.4, Problem 10QR

(a)

To determine

To find: The value of $a$ for which the equation $\sqrt{a^2}=a$ is true.

Answer to Problem 10QR

The value of $a$ for which $\sqrt{a^2}=a$ is true is $a\ge 0$ .

Explanation of Solution

Given information: The equation is $\sqrt{a^2}=a$ .

Calculation:

The equation $\sqrt{a^2}=a$ is only true for $a\ge 0$ .

Therefore, the value of $a$ for which $\sqrt{a^2}=a$ is true is $a\ge 0$ .

(b)

To determine

To find: The value of $a$ for which the equation $\sqrt{a^2}=\pm a$ is true.

Answer to Problem 10QR

The equation $\sqrt{a^2}=\pm a$ is true for all the real values of $a$ .

Explanation of Solution

Given information: The equation is $\sqrt{a^2}=\pm a$ .

Calculation:

The equation $\sqrt{a^2}=\pm a$ can be written as $\sqrt{a^2}=a$ and $\sqrt{a^2}=-a$ .

The equation $\sqrt{a^2}=a$ will be true for $a\ge 0$ and the equation $\sqrt{a^2}=-a$ will be true for $a\le 0$ .

So, at least one of the two equations is true for any real value of $a$ .

Therefore, the equation $\sqrt{a^2}=\pm a$ is true for all the real values of $a$ .

(c)

To determine

To find: The value of $a$ for which the equation $\sqrt{4a^2}=2\left|a\right|$ is true.

Answer to Problem 10QR

The equation $\sqrt{4a^2}=2\left|a\right|$ is true for all the real values of $a$ .

Explanation of Solution

Given information: The equation is $\sqrt{4a^2}=2\left|a\right|$ .

Calculation:

The equation $\sqrt{4a^2}=2\left|a\right|$ can be written as $2\sqrt{a^2}=2\left|a\right|$ or $2\left|a\right|=2\left|a\right|$ .

As the left hand side of the equation is equal to the right hand side of the equation.

So, at least one of the two equations is true for any real value of $a$ .

Therefore, the equation $\sqrt{4a^2}=2\left|a\right|$ is true for all the real values of $a$ .