Problem 38 - College Algebra

Question

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

Step-by-Step Solution

Answer

The result in standard form is \( -\frac{3}{8} + \frac{\sqrt{7} i}{8} \).

Step 1: Identify the Complex Number

Firstly, recognize that \( \sqrt{-28} \) is a complex number. The square root of a negative number is not real, but it can be expressed in terms of an imaginary number, i, where \( i^2 = -1 \). Hence, \( \sqrt{-28} = \sqrt{-1*28} = \sqrt{28} i \). The fraction now becomes \( \frac{-12+ \sqrt{28}i}{32} \)

Step 2: Simplify the Fraction

Now that the complex number is in the standard form, the fraction can be simplified by dividing both the real and the imaginary parts by the denominator, i.e., -12/32 and \( \sqrt{28}i/32 \)

Step 3: Put in Standard Form

After dividing, the fraction becomes \( -\frac{3}{8} + \frac{\sqrt{28} i}{32} \). After further simplifying, it becomes \( -\frac{3}{8} + \frac{\sqrt{7} i}{8} \). So, the standard form is \( -\frac{3}{8} + \frac{\sqrt{7} i}{8} \).

Key Concepts

Complex Number Operations

Working with complex numbers involves several operations that may seem challenging at first, but with practice, they become quite manageable. Complex numbers have both a real part and an imaginary part, and operations can be performed on these numbers just like we do with real numbers, but with a twist because they involve the imaginary unit, \(i\).

There are some key operations:

• Addition and Subtraction: Add or subtract the real parts and the imaginary parts separately. For example, \((3 + 4i) + (1 + 2i) = 4 + 6i\).
• Multiplication: Use distributive property (similar to binomials), remembering that \(i^2 = -1\). For example, \((2 + 3i)(1 + 4i) = 2 + 8i + 3i + 12i^2 = 2 + 11i - 12 = -10 + 11i\).
• Division: This involves multiplying the numerator and the denominator by the conjugate of the denominator to eliminate \(i\) from the denominator.

Complex number operations extend into functions and expand our ability to solve equations involving square roots of negative numbers.

Standard Form of Complex Numbers

The standard form of a complex number is a key concept in understanding how these numbers are structured. A complex number is typically written as \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part.

Imagine the earlier exercise where we used the expression \(-\frac{3}{8} + \frac{\sqrt{7} i}{8}\):

• In this example, the real part is \(-\frac{3}{8}\).
• The imaginary part is \(\frac{\sqrt{7} i}{8}\).

Having both parts in this clear structure allows us to distinguish between the real and imaginary components easily. It is essential for performing operations like addition, subtraction, multiplication, or division. Being comfortable with this form is crucial since it simplifies computations and helps in solving various mathematical problems.

Imaginary Numbers

Imaginary numbers are fascinating and sometimes confusing. They arise when you take the square root of a negative number. The core of imaginary numbers is the imaginary unit \(i\), defined by \(i^2 = -1\). This property is what differentiates imaginary numbers from real numbers.

For example, in the given problem, \(\sqrt{-28}\) is an imaginary number because you are trying to find the square root of a negative. It can be expressed as \(\sqrt{-1 \times 28} = \sqrt{28}i\).

Imaginary numbers are not "unreal," but rather they extend our number system to solve equations that wouldn’t have solutions in the real number system alone. When combined with real numbers, they form complex numbers, opening up new possibilities in science, engineering, and mathematics.