Problem 49 - Precalculus Student Solutions Manual 5th

Question

Find each sum or difference. Write the answer in standard form. $$-i \sqrt{2}-2-(6-4 i \sqrt{2})-(5-i \sqrt{2})$$

Step-by-Step Solution

Answer

-13 + 4i\sqrt{2}

Step 1 - Simplify the expression inside the parentheses

First, distribute the negative sign to each term inside the parentheses: Simplify paragraphs: instructions Instead \(6-4i\sqrt{2}\) becomes \(-6+4i\sqrt{2}\). \(5-i\sqrt{2}\) becomes \(-5+i\sqrt{2}\).

Step 2 - Combine like terms

Rewrite the equation considering the results from Step 1: \(-i\sqrt{2}-2-6+4i\sqrt{2}-5+i\sqrt{2}\)Now combine the real and imaginary parts separately:Real parts: \(-2 - 6 - 5\)Imaginary parts: \(-i\sqrt{2} + 4i\sqrt{2} + i\sqrt{2}\)

Step 3 - Calculate the real part

Calculate the sum of the real parts: \(-2 - 6 - 5 = -13\)

Step 4 - Calculate the imaginary part

Calculate the sum of the imaginary parts: \(-i\sqrt{2}+4i\sqrt{2}+i\sqrt{2} = 4i\sqrt{2}\)Simplify:\(4i\sqrt{2}\)

Step 5 - Write the final answer in standard form

Combine the results from Step 3 and Step 4 to write the answer in standard form: \(-13 + 4i\sqrt{2}\)

Key Concepts

Imaginary Numbers

Imaginary numbers are numbers that can be written as a real number multiplied by the imaginary unit, denoted as \(i\). The imaginary unit \(i\) is defined by the property that \(i^2 = -1\). For example, \(i\text{\text{ or }} 3i\) are both imaginary numbers. These numbers enable us to extend the concept of the square root and solve equations that do not have real solutions. Understanding imaginary numbers is essential for working with complex numbers, where we often encounter components that include \(i\).

Real Parts

The real part of a complex number is the component that does not involve the imaginary unit \(i\). In a general complex number, \(a + bi\), \(a\) is the real part, and \(bi\) is the imaginary part. Let's see an example: in the complex number \(4 + 5i\), \(4\) is the real part. When we perform operations with complex numbers, identifying the real part helps us in combining like terms correctly. This means you add or subtract the real components separate from the imaginary ones.

Standard Form

The standard form of a complex number is written as \(a + bi\), where \(a\) is the real part, and \(bi\) is the imaginary part. Writing complex numbers in standard form makes it easier to perform arithmetical operations and compare values. For instance, if our final answer in the problem is \(-13 + 4i\text\text(sqrt{2}\text\text)\), we clearly see \(-13\) as the real part, and \(4i\text\text(sqrt{2})\text\) as the imaginary part. By positioning the real parts and imaginary parts properly, we maintain clarity and ensure our results are easily understood.

Combining Like Terms

Combining like terms is an essential algebraic process where we add or subtract terms that have the same variables or components. In the context of complex numbers, we combine the real parts separately from the imaginary parts. In our given problem, \(-i\sqrt{2} - 2 - (6 - 4i\sqrt{2}) - (5 - i\sqrt{2})\), we distribute the negative signs and rewrite terms: \(-i\sqrt{2} - 2 - 6 + 4i\sqrt{2} - 5 + i\sqrt{2}\). Next, we combine: Real parts: \(-2 - 6 - 5 = -13\), Imaginary parts: \(-i\sqrt{2} + 4i\sqrt{2} + i\sqrt{2} = 4i\sqrt{2}\). Finally, ensuring the standard form, we get \(-13 + 4i\sqrt{2}\).