Simplify. Write your response in $a+b i$ form.
$10 i(-7+17 i)=$
Answer (1)
Distribute 10 i to both terms inside the parentheses: 10 i × ( − 7 ) + 10 i × ( 17 i ) .
Simplify each term: − 70 i + 170 i 2 .
Substitute i 2 = − 1 : − 70 i + 170 ( − 1 ) = − 70 i − 170 .
Rewrite in the standard form a + bi : − 170 − 70 i .
Explanation
Understanding the Problem We are asked to simplify the expression 10 i ( − 7 + 17 i ) and write the result in the form a + bi , where a and b are real numbers.
Distributing First, distribute 10 i to both terms inside the parentheses: 10 i ( − 7 + 17 i ) = 10 i × ( − 7 ) + 10 i × ( 17 i )
Simplifying Terms Simplify each term: 10 i × ( − 7 ) = − 70 i 10 i × ( 17 i ) = 170 i 2 So we have: − 70 i + 170 i 2
Substituting i 2 = − 1 Recall that i 2 = − 1 , so substitute − 1 for i 2 :
− 70 i + 170 ( − 1 ) = − 70 i − 170
Final Answer Rewrite in the standard form a + bi :
− 170 − 70 i Thus, the simplified expression is − 170 − 70 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, if the impedance is − 7 + 17 i and the current is 10 i , the voltage can be calculated by multiplying these two complex numbers, resulting in − 170 − 70 i .
