To find the square root of [tex]\(-16\)[/tex], we need to understand what taking the square root of a negative number entails.
### Step-by-Step Solution:
1. Understanding Imaginary Numbers:
The square root of a negative number involves an imaginary unit, denoted as [tex]\( i \)[/tex], where [tex]\( i = \sqrt{-1} \)[/tex]. This concept allows us to work with square roots of negative numbers.
2. Simplify the Problem:
We need to express [tex]\(-16\)[/tex] in terms of the imaginary unit [tex]\( i \)[/tex]:
[tex]\[
-16 = 16 \cdot (-1)
\][/tex]
3. Apply the Square Root Property:
The square root of a product is the product of the square roots:
[tex]\[
\sqrt{-16} = \sqrt{16 \cdot (-1)} = \sqrt{16} \cdot \sqrt{-1}
\][/tex]
4. Simplify Both Parts Separately:
- The square root of [tex]\( 16 \)[/tex] is [tex]\( 4 \)[/tex], since [tex]\( 4^2 = 16 \)[/tex].
- The square root of [tex]\( -1 \)[/tex] is [tex]\( i \)[/tex], since by definition, [tex]\( i = \sqrt{-1} \)[/tex].
5. Combine Results:
Combining these results, we get:
[tex]\[
\sqrt{-16} = 4 \cdot i = 4i
\][/tex]
### Conclusion:
Thus, the square root of [tex]\(-16\)[/tex] is [tex]\( 4i \)[/tex]. Looking at the given options:
- [tex]\( -8i \)[/tex]
- [tex]\( -4i \)[/tex]
- [tex]\( 4i \)[/tex]
- [tex]\( 8i \)[/tex]
The correct answer is:
[tex]\[ 4i \][/tex]
