Question

perform the indicated operations and write the result in standard form. \\(\frac{-12 + \sqrt{-45}}{24}\\) \\(\frac{-12 + \sqrt{-45}}{24} = \square\\) (type an exact answer, using radicals as needed. type your answer in the form \\(a + bi\\).)

Explanation:

Step1: Simplify the imaginary radical

$\sqrt{-45} = \sqrt{45} \cdot i = \sqrt{9 \times 5} \cdot i = 3\sqrt{5}i$

Step2: Split the fraction into two terms

$\frac{-12 + 3\sqrt{5}i}{24} = \frac{-12}{24} + \frac{3\sqrt{5}i}{24}$

Step3: Simplify each fraction

$\frac{-12}{24} = -\frac{1}{2}$, $\frac{3\sqrt{5}i}{24} = \frac{\sqrt{5}}{8}i$

Answer:

$-\frac{1}{2} + \frac{\sqrt{5}}{8}i$