Problem 18 - College Algebra

Question

In Exercises \(9-20,\) find each product and write the result in standard form. $$(-7-4 i)(-7+4 i)$$

Step-by-Step Solution

Answer

The product of the given complex numbers is 65.

Step 1: Multiply the Real Parts

First, multiply the real parts of both complex numbers: \((-7) \times (-7) = 49\)

Step 2: Multiply the Imaginary Parts

Next, multiply the imaginary parts of both complex numbers: \((-4i) \times (4i) = -16i^2\). Remembering that \(i^2 = -1\), this simplifies to \(-16 \times -1 = 16\)

Step 3: Combine the Results

Add the products from Steps 1 and 2 to get the final result. Remember that the imaginary part contributes to the real part of the product as a consequence of \(i^2 = -1\). So, the answer will be \(49 + 16 = 65\).

Key Concepts

Standard Form

In mathematics, the standard form of a complex number is an essential concept for working with these unique numbers. Complex numbers consist of two parts: a real part and an imaginary part. The standard form is presented as \(a + bi\), where \(a\) is the real component and \(bi\) is the imaginary component, with \(b\) being a real number factor of the imaginary unit \(i\). This clear and consistent format allows for straightforward calculations and representations of complex numbers in algebraic expressions.

• The real part, \(a\), is similar to the regular numbers we're used to handling, such as integers or fractions.
• The imaginary part, \(bi\), incorporates the imaginary unit \(i\), which distinguishes complex numbers from regular real numbers.
• The combination of both parts, \(a + bi\), creates a number system capable of expressing a broader range of solutions, particularly important in solving quadratic equations.

Imaginary Unit

The imaginary unit, denoted as \(i\), plays a fundamental role in the arithmetic of complex numbers. Unlike real numbers, \(i\) embodies the square root of \(-1\), allowing for mathematical expressions that can't be achieved using only real numbers. The definition is vital for understanding imaginary numbers, as it permits association with real coefficients. This aids in evaluating otherwise impossible equations in real numbers. Here are some key characteristics:

• \(i^2 = -1\), which is the defining property of the imaginary unit.
• Higher powers of \(i\) cycle in a predictable manner: \(i^3 = -i\), \(i^4 = 1\), and continue repeating every four powers.
• Multiplyingby \(i\) transforms real numbers into imaginary, and vice versa when dividing by \(i\), in various algebraic manipulations.

Product of Complex Numbers

The operation of multiplying complex numbers involves a few steps but follows a systematic approach similar to multiplying binomials. Considering two complex numbers \((a+bi)\) and \((c+di)\), their product is calculated using the distributive property:

• First, multiply the real parts: \(a \times c\).
• Next, multiply each real part by the opposite imaginary part and vice versa: \(a \times di + bi \times c\).
• Finally, multiply the imaginary parts together: \(bi \times di\), remembering that \(i^2 = -1\). This step contributes purely as a real number due to \(i^2\).
• Sum all calculated terms together to form the resultant complex number in standard form.

To illustrate, when multiplying the complex pair \((-7-4i)\) and \((-7+4i)\), each step falls in place to eventually yield the real number 65 in standard form as the imaginary parts cancel out.