Simplify. Write your response in $a+b i$ form.
$\sqrt{-9} \cdot \sqrt{-4}=$
Answer (2)
Rewrite − 9 as 3 i .
Rewrite − 4 as 2 i .
Multiply the two expressions: ( 3 i ) ( 2 i ) = 6 i 2 .
Substitute i 2 = − 1 to get 6 ( − 1 ) = − 6 , which is − 6 + 0 i .
The simplified form is − 6 + 0 i .
Explanation
Understanding the problem We are asked to simplify the expression − 9 ⋅ − 4 and write the result in the form a + bi , where a and b are real numbers.
Simplifying the first term First, we rewrite − 9 using the imaginary unit i , where i = − 1 . Thus, − 9 = 9 ⋅ − 1 = 9 ⋅ − 1 = 3 i .
Simplifying the second term Next, we rewrite − 4 using the imaginary unit i . Thus, − 4 = 4 ⋅ − 1 = 4 ⋅ − 1 = 2 i .
Multiplying the terms Now, we multiply the two simplified expressions: ( 3 i ) ( 2 i ) = 3 ⋅ 2 ⋅ i ⋅ i = 6 i 2 .
Substituting i^2 = -1 Since i = − 1 , we have i 2 = − 1 . Substituting this into our expression, we get 6 i 2 = 6 ( − 1 ) = − 6 .
Writing in a+bi form Finally, we write the result in the form a + bi . Since − 6 is a real number, we can write it as − 6 + 0 i .
Examples
Complex numbers are used in electrical engineering to analyze alternating current circuits. The impedance of a circuit, which includes resistance and reactance, is represented using complex numbers. Simplifying expressions involving square roots of negative numbers is essential in this field to determine circuit behavior and design efficient systems. For example, calculating the total impedance in a series circuit involves adding complex impedances, which often requires simplifying expressions like the one in this problem.
The expression − 9 ⋅ − 4 simplifies to − 6 + 0 i when expressed in the form of a + bi . This is derived by recognizing that − 9 = 3 i and − 4 = 2 i , and subsequently multiplying the two results. The final answer is − 6 + 0 i .
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