To express [tex]\(\sqrt{-36}\)[/tex] in terms of [tex]\(i\)[/tex], we need to consider the definition of the imaginary unit [tex]\(i\)[/tex], which is defined as [tex]\(i = \sqrt{-1}\)[/tex].
Here's the step-by-step process:
1. Identify the negative under the square root:
The expression [tex]\(\sqrt{-36}\)[/tex] involves a negative number under the square root, which suggests the use of the imaginary unit [tex]\(i\)[/tex].
2. Rewrite the square root of the negative number:
Recognize that [tex]\(\sqrt{-36}\)[/tex] can be expressed as [tex]\(\sqrt{36 \cdot (-1)}\)[/tex].
3. Separate the product under the square root:
Using the property of square roots, [tex]\(\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}\)[/tex], we can separate the square root into two parts:
[tex]\[
\sqrt{36 \cdot (-1)} = \sqrt{36} \cdot \sqrt{-1}
\][/tex]
4. Simplify each component:
- [tex]\(\sqrt{36}\)[/tex] is simply 6 because 6 is the non-negative square root of 36.
- [tex]\(\sqrt{-1}\)[/tex] is defined as [tex]\(i\)[/tex].
5. Combine the results:
Therefore, [tex]\(\sqrt{36} \cdot \sqrt{-1}\)[/tex] becomes:
[tex]\[
6 \cdot i
\][/tex]
So, the expression [tex]\(\sqrt{-36}\)[/tex] in terms of [tex]\(i\)[/tex] is:
[tex]\[
6i
\][/tex]
