Questions: Consider the expression -√(-a). Decide whether it is positive, negative, 0, or not a real number in each case. (a) a>0 (b) a<0 (c) a=0

Transcript text: Consider the expression $-\sqrt{-a}$. Decide whether it is positive, negative, 0 , or not a real number in each case. (a) $a>0$ (b) $a<0$ (c) $a=0$

Solution

Solution Steps

To determine the nature of the expression $-\sqrt{-a}$, we need to consider the value of $a$ in each case:

(a) If $a > 0$, then $-a$ is negative, and the square root of a negative number is not a real number.

(b) If $a < 0$, then $-a$ is positive, and the square root of a positive number is real. The negative sign outside the square root makes the expression negative.

Step 1: Evaluate for $ a > 0 $

When $ a > 0 $, we have $ -a < 0 $. Therefore, the expression $ -\sqrt{-a} $ involves the square root of a negative number, which is not a real number. Thus, the result is:

\[ \text{Result for } a > 0: \text{not a real number} \]

Step 2: Evaluate for $ a < 0 $

For $ a < 0 $, we find that $ -a > 0 $. Consequently, $ \sqrt{-a} $ is a positive real number. The expression becomes $ -\sqrt{-a} < 0 $, indicating that the result is negative. Therefore, we have:

\[ \text{Result for } a < 0: \text{negative} \]

Step 3: Evaluate for $ a = 0 $

When $ a = 0 $, we have $ -a = 0 $. Thus, the expression simplifies to $ -\sqrt{0} = 0 $. Therefore, the result is:

\[ \text{Result for } a = 0: 0 \]