Simplify, if possible.
$-1-\sqrt{-12}=$
(Simplify your answer. Type an exact answer, using radicals and $i$ as needed.)
Answer (1)
Rewrite the square root of the negative number using the imaginary unit: − 12 = 12 ⋅ − 1 .
Simplify the square root: 12 = 2 3 .
Substitute − 1 with i .
Combine the terms to get the final simplified expression: − 1 − 2 i 3 .
Explanation
Understanding the Problem We are asked to simplify the expression − 1 − − 12 . This involves simplifying a square root of a negative number, which will introduce the imaginary unit i .
Separating the Radical First, let's rewrite the square root of the negative number as a product of the square root of its positive counterpart and − 1 : − 12 = 12 ⋅ − 1
Simplifying the Square Root Now, we simplify 12 . We look for perfect square factors of 12. Since 12 = 4 ⋅ 3 , we have: 12 = 4 ⋅ 3 = 4 ⋅ 3 = 2 3
Introducing the Imaginary Unit Recall that − 1 is defined as the imaginary unit i . So, we can replace − 1 with i :
− 1 = i
Final Simplification Now, substitute the simplified radical back into the original expression: − 1 − − 12 = − 1 − ( 2 3 i ) = − 1 − 2 i 3
Examples
Complex numbers, which include imaginary numbers, are used in electrical engineering to analyze alternating current circuits. They help in representing and calculating impedance, which is the opposition to the flow of current in an AC circuit. Imaginary numbers are also crucial in quantum mechanics, where they are used to describe the wave functions of particles.
