- First, simplify individual radical terms to get \(3, 2i, 24,\) and \(8i\).
- Substitute these into the expression to yield \((-3+2i) - (48+8i)\).
- Distribute the negative and combine terms to find the simplified value of \(-51-6i\).
- Identify the equivalent options as \(-3+2 i -2(24)-8 i\) and \(-51-6 i\).
- Final Answer: \(\boxed{\text{-3+2i - 2(24) - 8i, -51-6i}}\)
Explanation
- Simplifying the Radical Terms
We are given the complex expression \((-\sqrt{9}+\sqrt{-4})-(2 \sqrt{576}+\sqrt{-64})\) and asked to identify which of the provided options are equivalent to its simplified form. To simplify, we first address each radical term individually, keeping in mind the definition of the imaginary unit \(i = \sqrt{-1}\).
- \(\sqrt{9} = 3\)
- \(\sqrt{-4} = \sqrt{4 \times -1} = 2i\)
- \(\sqrt{576} = 24\)
- \(\sqrt{-64} = \sqrt{64 \times -1} = 8i\)
- Simplifying the Full Expression
Now, substitute these values back into the original expression:
\[E = (-3 + 2i) - (2 \times 24 + 8i)\]
By distributing the negative sign, we obtain:
\[E = -3 + 2i - 48 - 8i\]
Combining the real parts (\(-3 - 48 = -51\)) and the imaginary parts (\(2i - 8i = -6i\)), we get the simplified standard form:
\[E = -51 - 6i\]
- Evaluating the Options
We compare the simplified form \(E = -51 - 6i\) with each of the provided options:
- Option 1: \(-3-2 i -2(24)+8 i = -3 - 2i - 48 + 8i = -51 + 6i\) (Incorrect)
- Option 2: \(-3+2 i -2(24)-8 i = -3 + 2i - 48 - 8i = -51 - 6i\) (Correct)
- Option 3: \(-51-6 i\) (Correct)
- Option 4: \(-51+6 i\) (Incorrect)
- Option 5: \(45+10 i\) (Incorrect)
- Option 6: \(-3+2 i +2(24)+8 i = -3 + 2i + 48 + 8i = 45 + 10i\) (Incorrect)
The equivalent expressions are \(-3+2 i -2(24)-8 i\) and \(-51-6 i\).
Examples
Understanding complex number arithmetic is essential in fields like electrical engineering, where alternating current circuits are analyzed using phasors. Just as we simplified the expression here by combining real and imaginary components, engineers simplify complex impedance equations to design efficient power systems and signal processing filters. Mastering these fundamentals allows for the precise calculation and manipulation of oscillatory systems in technology and physics.