Problem 40 - College Algebra and Calculus: An Applied Approach

Question

Perform the indicated operation and write the result in standard form. $$(3-7 i)^{2}$$

Step-by-Step Solution

Answer

The result of squaring the given complex number is \(58 - 42i\).

Step 1: Expanding the Squared Expression

Expanding the expression by applying the formula \((a - b)^2 = a^2 - 2ab + b^2\), the expression becomes \( (3 - 7i)^2 = 3^2 - 2 * 3 * 7i + (7i)^2\).

Step 2: Simplifying the Expression

Simplify the expression by performing the multiplications and the square: \( 9 - 42i - 49 \cdot i^2 \). Remember that \(i^2 = -1\), so the expression becomes \( 9 - 42i + 49\).

Step 3: Gathering real and imaginary terms

The expression above contains both real numbers and imaginary numbers. Collect the real numbers and the coefficients of the imaginary number separately to get the standard form. After adding 9 and 49, this results in \(58 - 42i\).

Key Concepts

Operations with Complex Numbers

Complex numbers often appear in the form \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part, with \(i\) representing the imaginary unit \(\sqrt{-1}\). There are several basic operations that can be performed on complex numbers, including addition, subtraction, multiplication, and division.

When performing operations like multiplication or raising to a power, it's essential to apply algebraic rules, bearing in mind the property \(i^2 = -1\). For example, using the formula \((a - b)^2 = a^2 - 2ab + b^2\), we can expand expressions like \((3 - 7i)^2\) by multiplying each part and substituting \(i^2 = -1\) to simplify the calculations further.

• Multiply by distributing each term carefully.
• Use \(i^2 = -1\) to remove \(i^2\) terms.
• Combine like terms to simplify.

By practicing these steps, performing operations on complex numbers becomes a more intuitive and easier task.

Complex Conjugate

The concept of the complex conjugate is crucial in simplifying complex number expressions, especially when dividing or rationalizing denominators. The conjugate of a complex number \(a + bi\) is \(a - bi\).

Multiplying a complex number by its conjugate results in a real number because the imaginary parts cancel each other out, leaving \(a^2 + b^2\). This property is especially useful for removing imaginary numbers from denominators.

• Use the conjugate to simplify division operations.
• The product of a complex number and its conjugate is always a positive real number if \(b eq 0\).
• Recognizing conjugates can aid in simplifying expressions involving complex numbers.

Understanding and using the complex conjugate helps to transform complex expressions into simpler forms, facilitating more straightforward calculations.

Standard Form of Complex Numbers

Complex numbers are conventionally represented in their standard form as \(a + bi\), where \(a\) and \(b\) are real numbers. This representation is helpful for easily identifying the real and imaginary components of the number.

After performing operations on complex numbers, presenting the result in standard form is vital as it clearly delineates these two components. For example, simplifying \((3 - 7i)^2\) produces \(58 - 42i\), which is already in standard form:

• Real part: 58
• Imaginary part: -42

Ensuring results are in standard form after calculations makes interpreting them easier. This practice becomes second nature as you become more familiar with complex numbers, improving both clarity and accuracy in your mathematical work.