To solve for the square root of \(-28\), we need to handle the square root of a negative number, which involves imaginary numbers. Here's the step-by-step solution:
1. Identify the negative aspect:
The number under the square root is \(-28\), which is negative. We know that the square root of a negative number involves the imaginary unit \(i\), where \(i = \sqrt{-1}\).
2. Express the negative number using \(i\):
Rewrite \(-28\) as \(-1 \times 28\).
3. Separate the components under the square root:
Using the property of square roots that \(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\), we have:
[tex]\[
\sqrt{-28} = \sqrt{-1 \times 28} = \sqrt{-1} \times \sqrt{28}
\][/tex]
4. Simplify the square root of \(-1\):
We know that \(\sqrt{-1} = i\).
5. Simplify the square root of \(28\):
The number \(28\) can be factored into its prime factors as \(28 = 4 \times 7\). Thus:
[tex]\[
\sqrt{28} = \sqrt{4 \times 7} = \sqrt{4} \times \sqrt{7}
\][/tex]
Since \(\sqrt{4} = 2\), this simplifies to:
[tex]\[
\sqrt{28} = 2\sqrt{7}
\][/tex]
6. Combine the results:
Now, substituting back into our expression, we have:
[tex]\[
\sqrt{-28} = \sqrt{-1} \times \sqrt{28} = i \times 2\sqrt{7}
\][/tex]
7. Write the final answer:
Simplifying the multiplication gives us:
[tex]\[
\sqrt{-28} = 2\sqrt{7}i
\][/tex]
Thus, the square root of [tex]\(-28\)[/tex] is [tex]\(2\sqrt{7}i\)[/tex].
