Problem 38 - Algebra and Trigonometry

Question

Perform the indicated operations and write the result in standard form. $$ \frac{-12+\sqrt{-28}}{32} $$

Step-by-Step Solution

Answer

-3/8 + sqrt(7)/16 * i

Step 1: Identify the Imaginary Part

Identify the imaginary part of the number. This involves recognizing that the square root of a negative number is imaginary. Therefore, \(\sqrt{-28}\) is imaginary. The imaginary number can be expressed in terms of i, where \(i^2 = -1\). Thus, \(\sqrt{-28}= \sqrt{28} * i\), as \(\sqrt{-1}=i\). Hence, \(\sqrt{-28}= 2\sqrt{7} * i\).

Step 2: Simplifying the Numerical Portion

Begin by simplifying the numerical part of the fraction. This means adding \(-12\) to \(2\sqrt{7} * i\) which results in \(-12 + 2\sqrt{7} * i\).

Step 3: Substituting in the Denominator

Now, replace the denominator in the given expression. With the numerator as \(-12 + 2\sqrt{7} * i\) and the denominator as \(32\), the expression becomes \(\frac{-12 + 2\sqrt{7} * i}{32}\).

Step 4: Division of Numerator by Denominator

Divide each term in the numerator by the denominator: \(\frac{-12}{32} + \frac{2\sqrt{7} * i}{32}\). This simplifies to \(-\frac{3}{8} + \frac{\sqrt{7} * i}{16}\).

Step 5: Writing in Standard Form

The standard form of a complex number is \(a + bi\) where \(a\) is the real part and \(bi\) is the imaginary part. Substitute \(a\) with \(-\frac{3}{8}\) and \(b\) with \(\frac{\sqrt{7}}{16}\) which gives us \(-\frac{3}{8} + \frac{\sqrt{7}}{16}i\). This is the required standard form of the given expression.