Perform the indicated operations and write the result in standard form.

StartFraction negative 12 plus StartRoot negative 18 EndRoot Over 60 EndFraction

The expression you are working with is:

\[
\frac{-12 + \sqrt{-18}}{60}
\]

### Step 1: Simplify the square root of a negative number
The square root of a negative number introduces an imaginary unit \(i\), where \(i = \sqrt{-1}\).

\[
\sqrt{-18} = \sqrt{18} \cdot \sqrt{-1} = \sqrt{18}i
\]

Now, simplify \(\sqrt{18}\):

\[
\sqrt{18} = \sqrt{9 \cdot 2} = 3\sqrt{2}
\]

Thus:

\[
\sqrt{-18} = 3\sqrt{2}i
\]

### Step 2: Substitute this result back into the original expression
Now substitute \(3\sqrt{2}i\) for \(\sqrt{-18}\) in the expression:

\[
\frac{-12 + 3\sqrt{2}i}{60}
\]

### Step 3: Simplify the fraction
Split the fraction into two parts:

\[
\frac{-12}{60} + \frac{3\sqrt{2}i}{60}
\]

Simplify each term:

\[
\frac{-12}{60} = -\frac{1}{5}
\]

\[
\frac{3\sqrt{2}i}{60} = \frac{\sqrt{2}i}{20}
\]

Thus, the expression in standard form is:

\[
-\frac{1}{5} + \frac{\sqrt{2}i}{20}
\]

### Final Answer:
The result in standard form is:

\[
-\frac{1}{5} + \frac{\sqrt{2}i}{20}
\]

Would you like more details on any part of this process?

Here are 5 related questions to expand on this:

1. How do you simplify complex numbers in different forms?
2. What are the steps to convert a complex fraction to standard form?
3. How do you interpret the imaginary unit \(i\) in more advanced math?
4. What properties of radicals help simplify square roots like \(\sqrt{18}\)?
5. What is the difference between real and imaginary parts of a complex number?

**Tip:** When simplifying complex numbers, always separate the real and imaginary parts for easier manipulation.

Perform the indicated operations and write the result in standard form. StartFraction negative 12 plus StartRoot negative 18 EndRoot Over 60 EndFraction

Learn how to simplify a complex number expression involving imaginary numbers and radicals. This solution walks you through simplifying sqrt(-18), breaking it into real and imaginary parts, and writing the result in standard form. Suitable for high school students, Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation