Select all the correct answers.

Which expressions are equivalent to the given expression?
[tex]\[
(-\sqrt{9}+\sqrt{-4})-(2 \sqrt{576}+\sqrt{-64})
\][/tex]

A. [tex]\(-51 - 6i\)[/tex]

B. [tex]\(-3 - 2i - 2(24) + 8i\)[/tex]

C. [tex]\(45 + 10i\)[/tex]

D. [tex]\(-3 + 2i + 2(24) + 8i\)[/tex]

E. [tex]\(-3 + 2i - 2(24) - 8i\)[/tex]

F. [tex]\(-51 + 6i\)[/tex]



Answer :

Let's break down the given expression step-by-step and simplify it. The given expression is:

[tex]\[ (-\sqrt{9}+\sqrt{-4})-(2 \sqrt{576}+\sqrt{-64}) \][/tex]

Step 1: Simplify each square root component.

[tex]\[ \sqrt{9} = 3 \][/tex]
[tex]\[ \sqrt{-4} = 2i \, (\text{since} \, \sqrt{-1} = i \, \text{and} \, \sqrt{4} = 2) \][/tex]
[tex]\[ 2 \sqrt{576} = 2 \times 24 = 48 \][/tex]
[tex]\[ \sqrt{-64} = 8i \, (\text{since} \, \sqrt{-1} = i \, \text{and} \, \sqrt{64} = 8) \][/tex]

Step 2: Substitute the simplified components back into the expression.
[tex]\[ (-3 + 2i) - (48 + 8i) \][/tex]

Step 3: Simplify further.

Combining the like terms:
[tex]\[ (-3 + 2i) - 48 - 8i \][/tex]
[tex]\[ -3 - 48 + 2i - 8i \][/tex]
[tex]\[ -51 - 6i \][/tex]

So, the simplified expression is:
[tex]\[ -51 - 6i \][/tex]

This matches our option, so the correct answers are:

[tex]\[ \boxed{-51-6i} \][/tex]

Now, checking other equivalent expressions:

1. [tex]\[ -51 - 6i \quad (\text{already determined as correct}) \][/tex]

2. [tex]\[ -3-2i-2(24)+8i \quad = \quad -3 - 2i - 48 + 8i \quad = \quad -51 + 6i \quad (\text{not correct}) \][/tex]

3. [tex]\[ 45 + 10i \quad (\text{not equivalent}) \][/tex]

4. [tex]\[ -3 + 2i + 2(24) + 8i \quad = \quad -3 + 2i + 48 + 8i \quad = \quad 45 + 10i \quad (\text{not correct}) \][/tex]

5. [tex]\[ -3 + 2i - 2(24) - 8i \quad = \quad -3 + 2i - 48 - 8i \quad = \quad -51 - 6i \quad (\text{correct}) \][/tex]

6. [tex]\[ -51 + 6i \quad (\text{not correct}) \][/tex]

Thus, the correct selections are:

[tex]\[ \boxed{-51-6i \quad \text{and} \quad -3+2i-2(24)-8i} \][/tex]