Perform the indicated operations and write the result in standard form. $\frac{-6 - \sqrt{-12}}{48}$

Verified step by step guidance

1. Identify the expression to simplify: \(\frac{-6 - \sqrt{-12}}{48}\).
2. Recognize that the square root of a negative number involves imaginary numbers. Rewrite \(\sqrt{-12}\) as \(\sqrt{12} \times \sqrt{-1}\), which is \(\sqrt{12}i\).
3. Simplify \(\sqrt{12}\) by factoring it into \(\sqrt{4 \times 3}\), which equals \$2\sqrt{3}\(. So, \)\sqrt{-12} = 2\sqrt{3}i$.
4. Substitute back into the original expression: \(\frac{-6 - 2\sqrt{3}i}{48}\).
5. Separate the real and imaginary parts by dividing both terms in the numerator by 48: \(\frac{-6}{48} - \frac{2\sqrt{3}i}{48}\). Then simplify each fraction to write the expression in standard form \(a + bi\).

Key Concepts

Complex Numbers and Imaginary Unit

Complex numbers include a real part and an imaginary part, where the imaginary unit 'i' is defined as √-1. Understanding how to express square roots of negative numbers using 'i' is essential for simplifying expressions like √-12.

Simplifying Radicals

Simplifying radicals involves factoring the number inside the square root to extract perfect squares. For example, √12 can be simplified to 2√3, which helps in rewriting expressions in a simpler form before performing operations.

Standard Form of a Complex Number

The standard form of a complex number is a + bi, where 'a' is the real part and 'b' is the coefficient of the imaginary part. Writing results in this form makes it easier to interpret and use complex numbers in further calculations.