Math Problem Statement

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

StartFraction negative 12 plus StartRoot negative 18 EndRoot Over 60 EndFraction

Solution

The expression you are working with is:

12+1860\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 ii, where i=1i = \sqrt{-1}.

18=181=18i\sqrt{-18} = \sqrt{18} \cdot \sqrt{-1} = \sqrt{18}i

Now, simplify 18\sqrt{18}:

18=92=32\sqrt{18} = \sqrt{9 \cdot 2} = 3\sqrt{2}

Thus:

18=32i\sqrt{-18} = 3\sqrt{2}i

Step 2: Substitute this result back into the original expression

Now substitute 32i3\sqrt{2}i for 18\sqrt{-18} in the expression:

12+32i60\frac{-12 + 3\sqrt{2}i}{60}

Step 3: Simplify the fraction

Split the fraction into two parts:

1260+32i60\frac{-12}{60} + \frac{3\sqrt{2}i}{60}

Simplify each term:

1260=15\frac{-12}{60} = -\frac{1}{5}

32i60=2i20\frac{3\sqrt{2}i}{60} = \frac{\sqrt{2}i}{20}

Thus, the expression in standard form is:

15+2i20-\frac{1}{5} + \frac{\sqrt{2}i}{20}

Final Answer:

The result in standard form is:

15+2i20-\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 ii in more advanced math?
  4. What properties of radicals help simplify square roots like 18\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.

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Math Problem Analysis

Mathematical Concepts

Complex Numbers
Simplifying Radicals
Imaginary Numbers
Fractions

Formulas

i = sqrt(-1)
Simplifying square roots: sqrt(a * b) = sqrt(a) * sqrt(b)

Theorems

Properties of Imaginary Numbers
Standard Form of Complex Numbers

Suitable Grade Level

Grades 9-12