Certainly! Let's solve the problem step-by-step.
We are given [tex]\( x \geq 0 \)[/tex] and the expression [tex]\( \sqrt{-x} = i\sqrt{x} \)[/tex], where [tex]\( i \)[/tex] is the imaginary unit, defined as [tex]\( i = \sqrt{-1} \)[/tex].
Now, we need to determine [tex]\( \sqrt{-5} \)[/tex] using the given information.
1. Identify [tex]\( x \)[/tex]:
Here, we have [tex]\( x = 5 \)[/tex].
2. Plug [tex]\( x \)[/tex] into the given formula:
According to the expression [tex]\( \sqrt{-x} = i\sqrt{x} \)[/tex],
[tex]\[
\sqrt{-5} = i \sqrt{5}
\][/tex]
3. Simplify the right-hand side:
To simplify [tex]\( i \sqrt{5} \)[/tex], we compute the square root of 5 and then multiply by the imaginary unit [tex]\( i \)[/tex].
4. Compute [tex]\( \sqrt{5} \)[/tex]:
The approximate value of [tex]\( \sqrt{5} \)[/tex] is [tex]\( 2.23606797749979 \)[/tex].
5. Multiply by [tex]\( i \)[/tex]:
Thus,
[tex]\[
\sqrt{-5} = i \cdot 2.23606797749979 = 2.23606797749979i
\][/tex]
So, the value of [tex]\( \sqrt{-5} \)[/tex] is approximately [tex]\( 2.23606797749979i \)[/tex].
