Select all the correct answers. Which expressions are equivalent to the given expression? [tex](-\sqrt{9}+\sqrt{-4})-(2 \sqrt{576}+\sqrt{-64})[/tex]
* -51-6i
* -3+2i -2(24)-8i
* 45+10i
* -51+6i
* -3+2i +2(24)+8i
* -3-2i-2(24)+8i
Answer (2)
Simplify the given expression by evaluating the square roots and combining real and imaginary terms.
The simplified expression is − 51 − 6 i .
Compare the simplified expression with the given options.
The equivalent expressions are − 51 − 6 i and − 3 + 2 i − 2 ( 24 ) − 8 i .
Explanation
Initial Analysis We are asked to find expressions equivalent to ( − 9 + − 4 ) − ( 2 576 + − 64 ) . Let's simplify this expression step by step. First, we evaluate the square roots.
Evaluating Square Roots We have 9 = 3 , − 4 = 2 i , 576 = 24 , and − 64 = 8 i . Substituting these values into the expression, we get: ( − 3 + 2 i ) − ( 2 ( 24 ) + 8 i )
Substituting Values Now, we simplify further: ( − 3 + 2 i ) − ( 48 + 8 i ) = − 3 + 2 i − 48 − 8 i
Simplifying the Expression Combining the real and imaginary parts, we have: − 3 − 48 + 2 i − 8 i = − 51 − 6 i
Comparing with Options Now, we compare the simplified expression − 51 − 6 i with the given options:
− 51 − 6 i (This matches our simplified expression)
− 3 + 2 i − 2 ( 24 ) − 8 i = − 3 + 2 i − 48 − 8 i = − 51 − 6 i (This matches our simplified expression)
45 + 10 i (This does not match)
− 51 + 6 i (This does not match)
− 3 + 2 i + 2 ( 24 ) + 8 i = − 3 + 2 i + 48 + 8 i = 45 + 10 i (This does not match)
− 3 − 2 i − 2 ( 24 ) + 8 i = − 3 − 2 i − 48 + 8 i = − 51 + 6 i (This does not match)
Final Answer Therefore, the expressions equivalent to the given expression are − 51 − 6 i and − 3 + 2 i − 2 ( 24 ) − 8 i .
Examples
Complex numbers might seem abstract, but they're incredibly useful in electrical engineering. Imagine designing a circuit where you need to analyze alternating current (AC). AC voltage and current can be represented as complex numbers, where the real part represents the resistance and the imaginary part represents the reactance (opposition to current change). By using complex numbers, engineers can easily calculate the impedance (total opposition to current flow) and analyze the circuit's behavior. This allows them to design efficient and stable electrical systems.
The expressions equivalent to the given expression are − 51 − 6 i and − 3 + 2 i − 2 ( 24 ) − 8 i .
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