To determine which expressions are equivalent to the given expression, let's break down the expression step-by-step:
Given expression:
[tex]\[ \left(-\sqrt{9} + \sqrt{-4}\right) - \left(2 \sqrt{576} + \sqrt{-64}\right) \][/tex]
First, let's evaluate each term separately:
1. [tex]\(\sqrt{9} = 3\)[/tex]
2. [tex]\(\sqrt{-4} = 2i\)[/tex] (since [tex]\(\sqrt{-4} = \sqrt{4 \cdot (-1)} = 2i\)[/tex])
3. [tex]\(\sqrt{576} = 24\)[/tex]
4. [tex]\(\sqrt{-64} = 8i\)[/tex] (since [tex]\(\sqrt{-64} = \sqrt{64 \cdot (-1)} = 8i\)[/tex])
Now, substitute these values into the expression:
[tex]\[ \left(-\sqrt{9} + \sqrt{-4}\right) - \left(2 \sqrt{576} + \sqrt{-64}\right) \][/tex]
This simplifies to:
[tex]\[ \left(-3 + 2i\right) - \left(2 \cdot 24 + 8i\right) \][/tex]
[tex]\[ -3 + 2i - 48 - 8i \][/tex]
Combine like terms:
[tex]\[ -3 - 48 + 2i - 8i \][/tex]
[tex]\[ -51 - 6i \][/tex]
Therefore, the given expression simplifies to:
[tex]\[ -51 - 6i \][/tex]
Let's compare this with each option provided to determine which ones are equivalent:
1. [tex]\(-3 + 2i + 2(24) + 8i\)[/tex]
[tex]\[ -3 + 2i + 48 + 8i \][/tex]
[tex]\[ 45 + 10i \][/tex]
2. [tex]\(-51 + 6i\)[/tex]
This does not equal [tex]\(-51 - 6i\)[/tex].
3. [tex]\(-51 - 6i\)[/tex]
This equals [tex]\(-51 - 6i\)[/tex].
4. [tex]\(45 + 10i\)[/tex]
This does not equal [tex]\(-51 - 6i\)[/tex].
5. [tex]\(-3 - 2i - 2(24) + 8i\)[/tex]
[tex]\[ -3 - 2i - 48 + 8i \][/tex]
[tex]\[ -51 + 6i \][/tex]
6. [tex]\(-3 + 2i - 2(24) - 8i\)[/tex]
[tex]\[ -3 + 2i - 48 - 8i \][/tex]
[tex]\[ -51 - 6i \][/tex]
Thus, the expressions equivalent to the given expression are:
[tex]\[ \boxed{3 \text{ and } 6} \][/tex]
