Simplify, if possible.
$-1-\sqrt{-8}=$
$\square$
(Simplify your answer. Type an exact answer, using radicals and i as needed.)
Answer (1)
Rewrite the square root of a negative number using the imaginary unit: − 8 = 8 ⋅ − 1 .
Simplify the radical: 8 = 2 2 .
Express in terms of i : − 8 = 2 2 i .
Substitute back into the original expression: − 1 − 2 2 i . The final answer is − 1 − 2 2 i .
Explanation
Understanding the Problem We are asked to simplify the expression − 1 − − 8 . This involves simplifying a square root of a negative number, which introduces imaginary numbers.
Rewriting the Radical First, we can rewrite − 8 as 8 × − 1 = 8 × − 1 .
Simplifying the Square Root of 8 Now, we simplify 8 . Since 8 = 4 × 2 , we have 8 = 4 × 2 = 4 × 2 = 2 2 .
Introducing the Imaginary Unit Recall that − 1 is defined as i , the imaginary unit. So, − 8 = 2 2 i .
Substituting Back into the Expression Finally, substitute this back into the original expression: − 1 − − 8 = − 1 − 2 2 i .
Final Answer Therefore, the simplified expression is − 1 − 2 2 i .
Examples
Imaginary numbers might seem abstract, but they're incredibly useful in electrical engineering. For example, when analyzing AC circuits, imaginary numbers help represent the phase difference between voltage and current. By using complex numbers (which combine real and imaginary parts), engineers can simplify calculations and design more efficient circuits. This is just one example of how imaginary numbers, which start as a mathematical concept, find practical applications in real-world engineering problems.
