Write in terms of $i$. Simplify your answer as much as possible. [tex]$\sqrt{-99}$[/tex]
Answer (1)
Rewrite − 99 as 99 ⋅ ( − 1 ) .
Separate the expression: 99 ⋅ − 1 .
Substitute − 1 with i : 99 i .
Simplify 99 to 3 11 , resulting in the final answer: 3 11 i .
Explanation
Understanding the problem We are asked to express − 99 in terms of i and simplify the result. Recall that i is the imaginary unit, defined as i = − 1 .
Rewriting the expression We can rewrite − 99 as 99 ⋅ ( − 1 ) .
Separating the terms Using the property ab = a ⋅ b , we can separate the expression into 99 ⋅ − 1 .
Introducing i Since i = − 1 , we can replace − 1 with i , so the expression becomes 99 i .
Simplifying the square root Now, we simplify 99 . We can factor 99 as 9 ⋅ 11 . Therefore, 99 = 9 ⋅ 11 .
Further simplification We can further simplify this as 9 ⋅ 11 = 9 ⋅ 11 = 3 11 .
Final result Substituting this back into our expression, we get 3 11 i .
Examples
Imaginary numbers might seem abstract, but they're essential in electrical engineering. For example, when analyzing alternating current (AC) circuits, imaginary numbers help represent the phase difference between voltage and current. This allows engineers to calculate impedance and design efficient power systems. Without imaginary numbers, AC circuit analysis would be much more complicated!
