Problem 41 - Precalculus Student Solutions Manual 5th

Question

Write each number in standard form a \(+b i\) $$\frac{-3+\sqrt{-18}}{24}$$

Step-by-Step Solution

Answer

\( \frac{-1}{8} + \frac{i\root{2}}{8} \)

Step 1: Simplify the Square Root

First, recognize that \(\root{-18}\) can be rewritten using imaginary numbers. We know \( \root{-18} = \root{-1} \times \root{18} = i \root{18}\). Then, simplify \( \root{18} \text{ which is } 3\root{2}\), giving \( \root{-18} = 3i\root{2}\).

Step 2: Substitute and Simplify

Replace \( \root{-18}\) in the original fraction with its simplified form \( 3i \root{2}\), we get \( \frac{-3 + 3i\root{2}}{24} \). Next, this fraction can be separated into its real and imaginary components: \( \frac{-3}{24} + \frac{3i\root{2}}{24} \).

Step 3: Further Simplify

Reduce the fractions. For the real part, \( \frac{-3}{24} = \frac{-1}{8}\). For the imaginary part, \( \frac{3i\root{2}}{24} = \frac{i\root{2}}{8}\). Therefore, the simplified form is \( \frac{-1}{8} + \frac{i\root{2}}{8} \).

Key Concepts

imaginary numbers

Imaginary numbers are a fundamental concept in mathematics, especially when dealing with complex numbers. The imaginary unit is denoted as 'i', which is defined as the square root of -1. This means that $$i^2 = -1.$$ When you encounter a negative number under a square root, you can rewrite it using 'i'. For example, $$\root{-18} = i \root{18}.$$ Imaginary numbers are essential in expressing solutions to equations that do not have real number solutions.

Imagine you need to solve $$x^2 + 1 = 0.$$ There are no real numbers that satisfy this equation because a square is always positive or zero. Using imaginary numbers, the solution becomes possible: $$x = i.$$ Imaginary numbers provide a way to expand our numerical understanding and solve more complex problems.

standard form

The standard form of a complex number is written as $$a + bi,$$ where 'a' is the real part and 'b' is the imaginary part. Complex numbers can be represented on a two-dimensional plane, known as the complex plane, with the real part 'a' plotted on the x-axis and the imaginary part 'bi' plotted on the y-axis.

To convert a given complex expression to standard form, you need to separate the real and imaginary components before simplifying each part independently. For example, consider $$\frac{-3 + \root{-18}}{24}.$$ By simplifying the square root and separating the terms, this can be rearranged and written as $$\frac{-1}{8} + \frac{i\root{2}}{8}.$$ This is now in the standard form 'a + bi'.

Knowing the standard form helps in performing various operations on complex numbers, such as addition, subtraction, and multiplication.

simplifying radicals

Simplifying radicals is an important skill when dealing with complex numbers. A radical expression is simplified when it has no square roots, cube roots, or higher order roots in the denominator and the radicand (the number inside the radical) does not contain any factors that are perfect squares, cubes, etc.

For example, to simplify \root{18}, you look for any perfect square factors within 18. Since 18 equals 2 times 9, and 9 is a perfect square (3^2), you can express \root{18} as \root{9 \times 2} = 3\root{2}.

When simplifying radicals in the realm of complex numbers, you often deal with negative radicands by incorporating the imaginary unit 'i'. For instance, \root{-18} can be rewritten using 'i' as $$i\root{18}.$$ Simplifying this, you get $$i \times 3\root{2} = 3i\root{2}.$$ This careful attention to breaking down and simplifying radicals ensures accurate and manageable expressions in further calculations.