Simplify the imaginary number $-\sqrt{-99}$.
Answer (1)
Rewrite − − 99 as − ( − 1 ) ( 99 ) .
Separate the terms: − − 1 99 .
Substitute − 1 with i : − i 99 .
Simplify 99 : − 3 i 11 .
The simplified form is − 3 i 11 .
Explanation
Understanding the Problem We are asked to simplify the imaginary number − − 99 . Let's break this down step by step.
Rewriting the Expression First, we recognize that the imaginary unit i is defined as − 1 . We can rewrite the given expression as follows: − − 99 = − ( − 1 ) ( 99 )
Separating the Terms Now, we use the property ab = a b to separate the terms: − ( − 1 ) ( 99 ) = − − 1 99
Substituting the Imaginary Unit Since − 1 = i , we substitute i into the expression: − i 99
Factoring 99 Next, we simplify 99 by factoring 99 into 9 × 11 :
− i 99 = − i 9 × 11
Separating the Factors Using the property ab = a b again, we get: − i 9 × 11 = − i 9 11
Simplifying the Square Root Since 9 = 3 , we have: − i ( 3 ) 11 = − 3 i 11
Final Answer Therefore, the simplified form of the imaginary number − − 99 is − 3 i 11 .
Examples
Imaginary numbers might seem abstract, but they're incredibly useful in electrical engineering. For example, when analyzing AC circuits, impedance (a measure of opposition to current) is often expressed using complex numbers, which include an imaginary part. Simplifying expressions with imaginary numbers, like we did here, helps engineers calculate and design efficient circuits. This ensures that devices like smartphones and computers receive the correct power and operate reliably.
