Certainly! Let's break this problem down step-by-step.
The given expression is:
[tex]\[
(-\sqrt{9} + \sqrt{-4}) - (2 \sqrt{576} + \sqrt{-64})
\][/tex]
Firstly, let's simplify each of the terms inside the expression:
1. [tex]\(-\sqrt{9}\)[/tex]:
[tex]\[
\sqrt{9} = 3 \implies -\sqrt{9} = -3
\][/tex]
2. [tex]\(\sqrt{-4}\)[/tex]:
[tex]\[
\sqrt{-4} = \sqrt{4} \cdot \sqrt{-1} = 2i
\][/tex]
3. [tex]\(2 \sqrt{576}\)[/tex]:
[tex]\[
\sqrt{576} = 24 \implies 2 \sqrt{576} = 2 \cdot 24 = 48
\][/tex]
4. [tex]\(\sqrt{-64}\)[/tex]:
[tex]\[
\sqrt{-64} = \sqrt{64} \cdot \sqrt{-1} = 8i
\][/tex]
Now, substituting these simplified terms back into the original expression:
[tex]\[
(-3 + 2i) - (48 + 8i)
\][/tex]
Next, let's perform the subtraction:
Real parts: [tex]\(-3 - 48 = -51\)[/tex]
Imaginary parts: [tex]\(2i - 8i = -6i\)[/tex]
Thus, the simplified form of the given expression is:
[tex]\[
-51 - 6i
\][/tex]
We can now compare this result to the provided options to see which ones are equivalent.
The given options and their evaluations are:
1. [tex]\(-3 + 2i + 2(24) + 8i\)[/tex]:
[tex]\[
= -3 + 2i + 48 + 8i = 45 + 10i
\][/tex]
Not equivalent.
2. [tex]\(45 + 10i\)[/tex]:
This is explicitly stated and is not equivalent to [tex]\(-51 - 6i\)[/tex].
3. [tex]\(-51 - 6i\)[/tex]:
This is exactly equivalent to our simplified result.
4. [tex]\(-3 + 2i - 2(24) - 8i\)[/tex]:
[tex]\[
= -3 + 2i - 48 - 8i = -3 - 48 + 2i - 8i = -51 - 6i
\][/tex]
Equivalent.
5. [tex]\(-3 - 2i - 2(24) + 8i\)[/tex]:
[tex]\[
= -3 - 2i - 48 + 8i = -3 - 48 + 6i = -51 + 6i
\][/tex]
Not equivalent.
6. [tex]\(-51 + 6i\)[/tex]:
This is not equivalent to [tex]\(-51 - 6i\)[/tex].
Hence, the correct equivalent expressions to the given expression are:
[tex]\[
-51 - 6i \quad \text{and} \quad -3 + 2i - 2(24) - 8i
\][/tex]
