To find the value of [tex]\(\sqrt{-64}\)[/tex], we need to understand the concept of complex numbers.
1. Understand the imaginary unit [tex]\(i\)[/tex]:
The imaginary unit [tex]\(i\)[/tex] is defined as [tex]\(\sqrt{-1}\)[/tex]. This is the foundation of complex numbers.
2. Express the negative under the square root using [tex]\(i\)[/tex]:
We can rewrite [tex]\(\sqrt{-64}\)[/tex] as [tex]\(\sqrt{64 \cdot -1}\)[/tex].
3. Separate the square roots:
We know that [tex]\(\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}\)[/tex]. Applying this property,
[tex]\[
\sqrt{64 \cdot -1} = \sqrt{64} \cdot \sqrt{-1}.
\][/tex]
4. Evaluate each square root:
- [tex]\(\sqrt{64}\)[/tex] is a real number. Since [tex]\(64\)[/tex] is a perfect square, [tex]\(\sqrt{64} = 8\)[/tex].
- [tex]\(\sqrt{-1}\)[/tex] by definition is [tex]\(i\)[/tex]. So, [tex]\(\sqrt{-1} = i\)[/tex].
5. Combine the results:
Multiplying the two results together, we get:
[tex]\[
\sqrt{64} \cdot \sqrt{-1} = 8 \cdot i.
\][/tex]
Hence, the value of [tex]\(\sqrt{-64}\)[/tex] is [tex]\(8i\)[/tex].
