Problem 35 - College Algebra and Calculus: An Applied Approach

Question

Perform the indicated operation and write the result in standard form. $$(3+4 i)(3-4 i)$$

Step-by-Step Solution

Answer

The result of the multiplication \((3+4i)(3-4i)\) is 25.

Step 1: Understand the formula for multiplication

First, it's important to understand how to multiply complex numbers. A complex number is of the form \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit. The formula for multiplying complex numbers is: \((a + bi)(c + di) = ac - bd + (ad + bc)i\).

Step 2: Apply the formula

Applying this formula to the provided exercise, \((3 + 4i)(3 - 4i)\), we get \((3*3 - 4*-4) + (3*-4 + 3*4)i\).

Step 3: Simplify the result

When we perform the operations in the parentheses, we get: \(9 + 16 + 0i\), which is equal to \(25 + 0i\).

Step 4: Write the result in standard form

The standard form of a complex number is \(a + bi\). The imaginary part of our result is 0, so we can omit it. Therefore, the result in standard form is 25.

Key Concepts

Multiplication of Complex Numbers

To multiply complex numbers, you need to follow a specific process that involves using the distributive property. Every complex number can be expressed as \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part. When dealing with the multiplication of two complex numbers, say \((a + bi)(c + di)\), the multiplication can be visualized as a two-step distribution:

• First, multiply each term in the first bracket by each term in the second bracket.
• Second, combine the like terms in the final expression.

The result will be \((ac - bd) + (ad + bc)i\). This formula arises from the arithmetic expansion where \(ac\) and \(-bd\) contribute to the real part, and \((ad + bc)\) contribute to the imaginary part.
Using this approach simplifies calculations and ensures that all components of the numbers are multiplied correctly.
In our example with \((3+4i)(3-4i)\), the result after using the formula is \(25 + 0i\).

Standard Form

The standard form of a complex number is critical for writing and understanding the outcome of any arithmetic operation involving complex numbers. It is written as \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit.
This form neatly separates the real and imaginary parts, making it easier to read and to perform additional calculations.
When you perform operations like addition, subtraction, multiplication, or division with complex numbers, your final result should always be expressed in this standard form. This makes sure that anyone can quickly identify the components of a complex number and understand its properties.
In our example, we simplified the result to \(25 + 0i\), which can simply be written as \(25\) because the imaginary part is zero.

Imaginary Unit

The imaginary unit \(i\) is fundamental to understanding complex numbers. It is defined by the property that \(i^2 = -1\). This is not something you encounter with real numbers and it's what allows complex numbers to represent solutions to equations that do not have real solutions.

• When dealing with expressions involving \(i\), like multiplication, it is important to remember that \(i^2\) simplifies to \(-1\).
• This characteristic is what turns certain terms imaginary when simplifying expressions.

In the multiplication example \((3+4i)(3-4i)\), understanding the effect of \(i^2\) is crucial in simplifying the expression correctly.
Notice when we multiplied \(-4i*4i\), it resulted in \(-16i^2\), which simplifies to \(16\) because \(i^2 = -1\).
This special property helps bridge complex numbers and allows them to be solved and expressed in a familiar, manageable form, especially when advocating the concept of complex conjugates used here.