Sure! Let's go step-by-step to find the solution for the given expression and the equivalent expressions listed.
The given expression is:
[tex]\[ (-\sqrt{9} + \sqrt{-4}) - (2\sqrt{576} + \sqrt{-64}) \][/tex]
1. First, simplify each of the square root terms:
- [tex]\(\sqrt{9} = 3\)[/tex]
- [tex]\(\sqrt{-4} = 2i\)[/tex]
- [tex]\(2\sqrt{576} = 2 \times 24 = 48\)[/tex]
- [tex]\(\sqrt{-64} = 8i\)[/tex]
2. Substitute these simplified values back into the expression:
[tex]\[ (-3 + 2i) - (48 + 8i) \][/tex]
3. Now simplify the expression by handling the subtraction separately for the real and imaginary parts:
- Real part: [tex]\(-3 - 48 = -51\)[/tex]
- Imaginary part: [tex]\(2i - 8i = -6i\)[/tex]
So, the simplified result is:
[tex]\[ -51 - 6i \][/tex]
Now, we need to determine which among the given expressions are equivalent to the simplified result [tex]\( -51 - 6i \)[/tex].
Let's examine the provided options one by one:
1. [tex]\( 45 + 10i \)[/tex]
- This is not equivalent to [tex]\( -51 - 6i \)[/tex].
2. [tex]\( -51 - 6i \)[/tex]
- This is directly equivalent to the simplified result [tex]\( -51 - 6i \)[/tex].
3. [tex]\( -3 + 2i + 2(24) + 8i \)[/tex]
- Simplify: [tex]\( -3 + 2i + 48 + 8i = 45 + 10i \)[/tex]
- This is not equivalent to [tex]\( -51 - 6i \)[/tex].
4. [tex]\( -51 + 6i \)[/tex]
- This is not equivalent to [tex]\( -51 - 6i \)[/tex].
5. [tex]\( -3 - 2i - 2(24) + 8i \)[/tex]
- Simplify: [tex]\( -3 - 2i - 48 + 8i = -51 + 6i \)[/tex]
- This is not equivalent to [tex]\( -51 - 6i \)[/tex].
6. [tex]\( -3 + 2i - 2(24) - 8i \)[/tex]
- Simplify: [tex]\( -3 + 2i - 48 - 8i = -51 - 6i \)[/tex]
- This is equivalent to [tex]\( -51 - 6i \)[/tex].
Thus, the correct and equivalent expressions to the given expression:
[tex]\[ (-\sqrt{9}+\sqrt{-4})-(2 \sqrt{576}+\sqrt{-64}) \][/tex]
are:
[tex]\[ -51 - 6i \text{ and } -3 + 2i - 2(24) - 8i \][/tex]
Therefore, the correct options are:
[tex]\[ \boxed{2 \text{ and } 6} \][/tex]
