Perform the indicated operations, and write the result in the form $a+b i$. (Simplify your answers completely.)
$(\sqrt{2}-\sqrt{-2})(\sqrt{32}+\sqrt{-2})$
Answer (2)
Rewrite the terms with imaginary units: − 2 = i 2 and 32 = 4 2 .
Substitute the rewritten terms into the expression: ( 2 − i 2 ) ( 4 2 + i 2 ) .
Expand the expression using the distributive property and simplify, remembering that i 2 = − 1 .
The simplified expression in the form a + bi is: 10 − 6 i .
Explanation
Understanding the Problem We are asked to perform the indicated operations and write the result in the form a + bi . The expression is ( 2 − − 2 ) ( 32 + − 2 ) . Our goal is to simplify this expression and express it in the standard form of a complex number.
Rewriting the Terms First, let's rewrite the terms involving the square root of negative numbers using the imaginary unit i , where i = − 1 . We have − 2 = 2 ⋅ − 1 = 2 ⋅ − 1 = i 2 and 32 = 16 ⋅ 2 = 4 2 .
Substituting the Values Now, substitute these values into the expression: ( 2 − i 2 ) ( 4 2 + i 2 )
Expanding the Expression Expand the expression using the distributive property (FOIL method):
( 2 − i 2 ) ( 4 2 + i 2 ) = ( 2 ) ( 4 2 ) + ( 2 ) ( i 2 ) + ( − i 2 ) ( 4 2 ) + ( − i 2 ) ( i 2 ) = 4 ( 2 ) 2 + i ( 2 ) 2 − 4 i ( 2 ) 2 − i 2 ( 2 ) 2 = 4 ( 2 ) + i ( 2 ) − 4 i ( 2 ) − ( − 1 ) ( 2 ) = 8 + 2 i − 8 i + 2 = 10 − 6 i
Final Answer The result is 10 − 6 i , which is already in the form a + bi , where a = 10 and b = − 6 .
Conclusion Therefore, the simplified expression is 10 − 6 i .
Examples
Complex numbers might seem abstract, but they're incredibly useful in electrical engineering. Imagine you're designing a circuit. The impedance, which is the opposition to the flow of alternating current, can be described using complex numbers. The real part represents resistance, and the imaginary part represents reactance. By performing operations with complex numbers, like we did in the problem, engineers can analyze and optimize circuits to ensure they function correctly. This helps in designing everything from smartphones to power grids.
To simplify the expression ( 2 − − 2 ) ( 32 + e wl in e − 2 ) , we rewrite it using imaginary units and expand it to get the final result as 10 − 6 i .
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