Simplify. Write your response in $a+b i$ form.
$-7 i(-13+17 i) =$
Answer (2)
Distribute − 7 i to both terms inside the parentheses: − 7 i × − 13 + ( − 7 i ) × 17 i .
Simplify the terms: 91 i − 119 i 2 .
Substitute i 2 = − 1 : 91 i − 119 ( − 1 ) = 91 i + 119 .
Write the expression in the standard form a + bi : 119 + 91 i .
Explanation
Understanding the Problem We are asked to simplify the expression − 7 i ( − 13 + 17 i ) and write the result in the form a + bi , where a and b are real numbers.
Distributing First, distribute − 7 i to both terms inside the parentheses: − 7 i × − 13 + ( − 7 i ) × 17 i
Simplifying Simplify the terms: 91 i − 119 i 2
Substituting i 2 = − 1 Recall that i 2 = − 1 . Substitute this into the expression: 91 i − 119 ( − 1 ) 91 i + 119
Standard Form Write the expression in the standard form a + bi :
119 + 91 i
Examples
Complex numbers are used in electrical engineering to analyze alternating current circuits. The impedance of a circuit, which is the opposition to the flow of current, is represented as a complex number. By simplifying expressions involving complex numbers, engineers can determine the voltage and current in a circuit, which is crucial for designing and troubleshooting electrical systems. For example, the expression − 7 i ( − 13 + 17 i ) could represent a simplified form of an impedance calculation in a circuit.
The expression − 7 i ( − 13 + 17 i ) simplifies to 119 + 91 i by distributing and using the fact that i 2 = − 1 . The final result is presented in standard complex number form. This process involves basic multiplication and substitution techniques in algebra.
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