Simplify. Write your response in $a+b i$ form.
$\sqrt{-169} \cdot \sqrt{-4} =$
Answer (1)
Rewrite the expression using the imaginary unit: − 169 ⋅ − 4 = 169 i ⋅ 4 i .
Simplify the square roots: 13 i ⋅ 2 i .
Multiply the terms: 26 i 2 .
Substitute i 2 = − 1 and simplify: 26 ( − 1 ) = − 26 . The final answer is − 26 .
Explanation
Understanding the Problem We are asked to simplify the expression − 169 ⋅ − 4 and write the result in the form a + bi , where a and b are real numbers. We need to remember that the imaginary unit i is defined as i = − 1 , which means i 2 = − 1 .
Rewriting the Expression First, we rewrite − 169 as 169 ⋅ − 1 and − 4 as 4 ⋅ − 1 .
Simplifying Square Roots Next, we simplify 169 to 13 and 4 to 2 . So we have 13 ⋅ − 1 and 2 ⋅ − 1 .
Introducing the Imaginary Unit Now, we replace − 1 with i , so the expression becomes ( 13 i ) ⋅ ( 2 i ) .
Multiplying the Terms We multiply the terms: 13 ⋅ 2 ⋅ i ⋅ i = 26 i 2 .
Substituting i 2 Since i 2 = − 1 , we substitute − 1 for i 2 : 26 ⋅ ( − 1 ) = − 26 .
Final Answer Finally, we write the result in a + bi form: − 26 + 0 i . Therefore, the simplified expression is − 26 + 0 i or simply − 26 .
Examples
Complex numbers, like the one we simplified, are used in electrical engineering to analyze alternating current circuits. The imaginary part represents the reactance, and the real part represents the resistance. By simplifying expressions with imaginary numbers, engineers can calculate the impedance and understand the behavior of circuits. This helps in designing efficient and stable electrical systems.
