Show all work to multiply $(2+\sqrt{-25})(4-\sqrt{-100})$.
Answer (2)
To multiply ( 2 + s q r t − 25 ) ( 4 − − 100 ) , we rewrite the square roots using the imaginary unit i and then apply the distributive property to expand the product. After simplifying, we find that the result is 58. The final answer in standard form is thus 58 + 0 i = 58 .
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Rewrite the square roots of negative numbers using the imaginary unit i : − 25 = 5 i and − 100 = 10 i .
Substitute these values into the expression: ( 2 + 5 i ) ( 4 − 10 i ) .
Expand the product using the distributive property: 8 − 20 i + 20 i − 50 i 2 .
Simplify, using i 2 = − 1 , to get the final result: 58 .
Explanation
Understanding the Problem We are asked to multiply two complex numbers: ( 2 + − 25 ) ( 4 − − 100 ) . Our goal is to express the result in the standard form a + bi , where a and b are real numbers.
Rewriting with Imaginary Unit First, we rewrite the square roots of negative numbers using the imaginary unit i , where i = − 1 . Thus, we have − 25 = 25 ⋅ − 1 = 5 i and − 100 = 100 ⋅ − 1 = 10 i .
Expanding the Product Now, substitute these values into the expression: ( 2 + 5 i ) ( 4 − 10 i ) . Next, we expand the product using the distributive property (also known as the FOIL method): ( 2 ) ( 4 ) + ( 2 ) ( − 10 i ) + ( 5 i ) ( 4 ) + ( 5 i ) ( − 10 i ) = 8 − 20 i + 20 i − 50 i 2 .
Simplifying the Expression Since i 2 = − 1 , we substitute this value into the expression: 8 − 20 i + 20 i − 50 ( − 1 ) = 8 − 20 i + 20 i + 50. Now, we simplify further by combining the real and imaginary parts: ( 8 + 50 ) + ( − 20 i + 20 i ) = 58 + 0 i .
Final Result Therefore, the final result in the standard form a + bi is: 58 + 0 i = 58. So, the product of the given complex numbers is simply 58.
Examples
Complex numbers might seem abstract, but they're incredibly useful in fields like electrical engineering. Imagine designing a circuit where you need to analyze alternating current (AC). AC voltage and current can be represented as complex numbers, making calculations much easier. By multiplying complex numbers, engineers can determine the impedance and phase shifts in circuits, ensuring efficient and stable performance. This helps in designing everything from power grids to smartphone chargers!
