Problem 15 - College Algebra Essentials

Question

In Exercises \(9-20,\) find each product and write the result in standard form. $$(3+5 i)(3-5 i)$$

Step-by-Step Solution

Answer

The product of the complex numbers \( (3+5i) \) and \( (3-5i) \) in standard form is -16.

Step 1: Expand the Product

First, distribute each term in the first parenthesis to each term in the second one. Thus the equation becomes: \((3+5i)(3) + (3+5i)(-5i).\)

Step 2: Simplify

In the next step, simplify the equation: \(3*3 - 3*5i + 5i*3 - 5i*5i \). Remember that \(i^2\) is equal to -1.

Step 3: Final result

Compute the multiplication respectively, which gives: \(9 - 15i + 15i - 25 \). The negative term in this equation is the result of the multiplication of \(i\) by itself, which is \(i^2 = -1\). Again, simplify the result by canceling out the equal magnitude terms: \(9-25 = -16\).

Key Concepts

Standard Form in Algebra

Understanding how to express a complex number in standard form is crucial when dealing with algebraic operations such as multiplication. The standard form for a complex number is typically written as \(a + bi\), where \(a\) represents the real part and \(b\) the imaginary component, with \(i\) being the imaginary unit. The beauty of this form lies in its simplicity; it streamlines the addition, subtraction, and multiplication of complex numbers.

When multiplying complex numbers, as in the exercise \( (3+5i)(3-5i) \), it's important to apply the distributive property just as you would with polynomials. An expanded product will often contain terms with \(i\), and your job is to simply combine like terms and reduce the expression whenever possible. After these steps, the final result should always be reorganized back into the standard form to make it easily readable and interpretable, in this case, yielding a purely real number.

Imaginary Numbers

Imaginary numbers, a fundamental concept for working with complex numbers, bear the unique property of being based on the square root of a negative number. Represented typically by \(i\), the imaginary unit, they account for solutions to equations that have no real-number solutions. For example, \(i^2 = -1\), is the defining property of the imaginary unit.

Understanding that \( i \)'s exponent patterns repeat every four powers provides a helpful shortcut in simplification. For instance, \(i^3 = i^2 * i = -1 * i = -i\), and \(i^4 = i^2 * i^2 = -1 * -1 = 1\). In the context of our example, the product \( -5i * 5i \) simplifies to \( -25 * i^2\), which then becomes \( -25 * -1 = 25 \), a real number. Thus, correctly handling imaginary numbers is essential for simplifying complex expressions.

Simplifying Expressions

Simplifying expressions, like the multiplication of complex numbers, requires careful attention to mathematical operations and properties. Simplification often includes expanding expressions, combining like terms, and reducing the expression by eliminating any terms that cancel out. This process often involves recognizing and applying known properties, like \(i^2 = -1\), effectively.

Consider our worked example \( (3+5i)(3-5i) \) once simplified yields \(9 - 15i + 15i - 25\). Notice the middle terms, \( -15i \) and \( 15i \) cancel each other out. What remains, the real numbers 9 and -25, sum up to \( -16\), leaving us with a simplified final answer in standard form with no imaginary component. Students should practice these steps to become proficient in simplifying complex expressions, thus expanding their algebraic capabilities.