Write the expression in terms of $i$ and simplify if possible.
$\sqrt{-108}=$ $\square$
Answer (1)
Rewrite − 108 as − 1 × 108 .
Substitute − 1 with i .
Simplify 108 to 6 3 .
Combine the results to get the final answer: 6 3 i .
Explanation
Understanding the problem We are asked to express − 108 in terms of i and simplify it. We know that i is the imaginary unit, defined as i = − 1 .
Rewriting the expression First, we can rewrite the expression as follows: − 108 = ( − 1 ) × ( 108 )
Separating the expression Using the property ab = a × b , we can separate the expression: − 108 = − 1 × 108
Substituting i Since i = − 1 , we can substitute i into the expression: − 108 = i × 108
Prime factorization of 108 Now, we need to simplify 108 . Let's find the prime factorization of 108: 108 = 2 × 54 = 2 × 2 × 27 = 2 2 × 3 × 9 = 2 2 × 3 × 3 × 3 = 2 2 × 3 3
Rewriting the square root So, we can rewrite 108 as: 108 = 2 2 × 3 3 = 2 2 × 3 2 × 3
Simplifying the square root Now, we simplify the square root: 2 2 × 3 2 × 3 = 2 2 × 3 2 × 3 = 2 × 3 × 3 = 6 3
Final answer Finally, we substitute this back into our expression: − 108 = i × 6 3 = 6 3 i
Examples
Complex numbers, involving the imaginary unit i , are used in electrical engineering to analyze AC circuits. They help represent the impedance, which is the opposition to the flow of current in an AC circuit. By using complex numbers, engineers can simplify the calculations and design more efficient circuits. For example, the voltage, current, and impedance in an AC circuit can be represented as complex numbers, allowing for easier analysis of the circuit's behavior.
