Answer :
To determine the equivalent expressions to the given expression [tex]\((-\sqrt{9}+\sqrt{-4})-(2 \sqrt{576}+\sqrt{-64})\)[/tex], let's simplify it step-by-step.
1. Simplify each term:
[tex]\[ \sqrt{9} = 3 \][/tex]
[tex]\[ \sqrt{-4} = 2i \quad (\text{since} \ \sqrt{-4} = \sqrt{4} \cdot \sqrt{-1} = 2i) \][/tex]
[tex]\[ 2 \sqrt{576} = 2 \cdot 24 = 48 \][/tex]
[tex]\[ \sqrt{-64} = 8i \quad (\text{since} \ \sqrt{-64} = \sqrt{64} \cdot \sqrt{-1} = 8i) \][/tex]
2. Substitute the simplified terms back into the expression:
[tex]\[ (-\sqrt{9}+\sqrt{-4})-(2 \sqrt{576}+\sqrt{-64}) = (-3 + 2i) - (48 + 8i) \][/tex]
3. Combine like terms:
[tex]\[ (-3 + 2i) - (48 + 8i) = -3 + 2i - 48 - 8i \][/tex]
Combining like terms for the real parts and the imaginary parts separately:
[tex]\[ (-3 - 48) + (2i - 8i) = -51 - 6i \][/tex]
Thus, the simplified form of the given expression is:
[tex]\[ -51 - 6i \][/tex]
Based on this simplification, the correct equivalent expression from the provided options is:
- [tex]\(-3 + 2i - 2(24) - 8i\)[/tex]
- [tex]\(-51 - 6i\)[/tex]
The full set of correct answers includes:
- [tex]\(-3 + 2i - 2(24) - 8i\)[/tex]
- [tex]\(-51 - 6i\)[/tex]
Thus, the correct selection of all equivalent expressions are:
[tex]\[ \boxed{-3+2 i-2(24)-8 i \text{ and } -51-6 i} \][/tex]
1. Simplify each term:
[tex]\[ \sqrt{9} = 3 \][/tex]
[tex]\[ \sqrt{-4} = 2i \quad (\text{since} \ \sqrt{-4} = \sqrt{4} \cdot \sqrt{-1} = 2i) \][/tex]
[tex]\[ 2 \sqrt{576} = 2 \cdot 24 = 48 \][/tex]
[tex]\[ \sqrt{-64} = 8i \quad (\text{since} \ \sqrt{-64} = \sqrt{64} \cdot \sqrt{-1} = 8i) \][/tex]
2. Substitute the simplified terms back into the expression:
[tex]\[ (-\sqrt{9}+\sqrt{-4})-(2 \sqrt{576}+\sqrt{-64}) = (-3 + 2i) - (48 + 8i) \][/tex]
3. Combine like terms:
[tex]\[ (-3 + 2i) - (48 + 8i) = -3 + 2i - 48 - 8i \][/tex]
Combining like terms for the real parts and the imaginary parts separately:
[tex]\[ (-3 - 48) + (2i - 8i) = -51 - 6i \][/tex]
Thus, the simplified form of the given expression is:
[tex]\[ -51 - 6i \][/tex]
Based on this simplification, the correct equivalent expression from the provided options is:
- [tex]\(-3 + 2i - 2(24) - 8i\)[/tex]
- [tex]\(-51 - 6i\)[/tex]
The full set of correct answers includes:
- [tex]\(-3 + 2i - 2(24) - 8i\)[/tex]
- [tex]\(-51 - 6i\)[/tex]
Thus, the correct selection of all equivalent expressions are:
[tex]\[ \boxed{-3+2 i-2(24)-8 i \text{ and } -51-6 i} \][/tex]
