To determine which choices are equivalent to the expression [tex]\(\sqrt{-36}\)[/tex], let's analyze the expression step-by-step:
1. The expression [tex]\(\sqrt{-36}\)[/tex] involves the square root of a negative number. To handle this, we use the imaginary unit [tex]\(i\)[/tex], where [tex]\(i = \sqrt{-1}\)[/tex].
2. We can rewrite [tex]\(\sqrt{-36}\)[/tex] using the properties of square roots and the imaginary unit:
[tex]\[
\sqrt{-36} = \sqrt{36 \cdot (-1)} = \sqrt{36} \cdot \sqrt{-1}
\][/tex]
3. We know:
[tex]\[
\sqrt{36} = 6 \quad \text{and} \quad \sqrt{-1} = i
\][/tex]
4. Substituting these values into the expression, we get:
[tex]\[
\sqrt{-36} = 6 \cdot i = 6i
\][/tex]
Now, let's check the choices:
A. [tex]\(i \sqrt{36}\)[/tex]:
[tex]\[
i \sqrt{36} = i \cdot 6 = 6i
\][/tex]
This expression is indeed equivalent to [tex]\(6i\)[/tex]. So, A is correct.
B. [tex]\(-\sqrt{36}\)[/tex]:
[tex]\[
-\sqrt{36} = -6
\][/tex]
This expression is just [tex]\(-6\)[/tex], which is not equivalent to [tex]\(6i\)[/tex]. So, B is incorrect.
C. [tex]\(-6\)[/tex]:
This is simply the number [tex]\(-6\)[/tex], which is not equivalent to [tex]\(6i\)[/tex]. So, C is incorrect.
D. [tex]\(6i\)[/tex]:
This is exactly the same as [tex]\(6i\)[/tex], which is what we found [tex]\(\sqrt{-36}\)[/tex] to be. So, D is correct.
Therefore, the choices that are equivalent to [tex]\(\sqrt{-36}\)[/tex] are:
[tex]\[
\boxed{1, 4}
\][/tex]
