Problem 13 - College Algebra Essentials

Question

In Exercises \(9-20,\) find each product and write the result in standard form. $$(7-5 i)(-2-3 i)$$

Step-by-Step Solution

Answer

The product of (7-5i) and (-2-3i) in standard form is \(-29 - 11i\).

Step 1: Distribute All Terms

Distribute all terms in the first parentheses to all terms in the second parentheses, including the imaginary components, following the distributive law of algebra. This results in \(7 * -2 + 7 * -3i - 5i * -2 - 5i * -3i \).

Step 2: Simplify Real Part

Simplify the terms without 'i' to get real part of the product. \(7 * -2 = - 14\). This is the real part.

Step 3: Simplify Imaginary Part

Simplify the terms with 'i' to get the imaginary part of the product. \(7 * -3i - 5i * -2 = -21i + 10i = -11i\). This is the imaginary part.

Step 4: Simplify the product of two imaginary numbers

Dealing with \( - 5i * -3i \), remember that \(i^2 = -1\). So, adding \( - 5i * -3i = 15i^2 = -15 \) to the real part.

Step 5: Combine Real and Imaginary Parts

Combine the real and imaginary parts to get the answer in standard form \(a + bi\). After considering all previous steps, the combined form is \(-14 -15 -11i = -29 - 11i\)

Key Concepts

Standard Form

Complex numbers can be intimidating at first, but they become much easier once you know how to express them in standard form. A complex number in standard form is written as \(a + bi\), where \(a\) represents the real part, and \(b\) represents the imaginary part. The letter \(i\) denotes the imaginary unit, which we will discuss later. Simplifying expressions into this form allows us to easily identify both the real and the imaginary components.

When you calculate the product of complex numbers and rearrange your results, always aim to express the final answer in standard form. This ensures clarity and consistency in all your mathematical computations.

Imaginary Unit

The imaginary unit, represented by \(i\), is a core element of complex numbers. The fundamental property of \(i\) is that \(i^2 = -1\). You can think of \(i\) as a tool that helps us solve mathematical problems involving square roots of negative numbers, which are not solvable within the real number system.

Understanding this property is crucial when working with complex numbers, especially during multiplication. For example, when multiplying two imaginary numbers like \(-5i\) and \(-3i\), remember to adjust for \(i^2 = -1\), turning \(15i^2\) into \(-15\). This conversion is necessary to properly handle imaginary components and find their contribution to the real part.

Distributive Law

The distributive law is a vital algebraic principle that helps in expanding expressions. When multiplying complex numbers, we use the distributive law to ensure each term from one set of parentheses is multiplied by each term from the other. For instance, given \((7-5i)\) and \((-2-3i)\), apply the distributive law as follows:

• Multiply \(7\) by \(-2\) and \(-3i\).
• Multiply \(-5i\) by \(-2\) and \(-3i\).

This results in \(7 \times -2 + 7 \times -3i - 5i \times -2 - 5i \times -3i\).

By distributing carefully, you can capture every part of both complex numbers, laying the groundwork for further simplification and correctly solving complex multiplication tasks.

Simplification

Simplification is the process of reducing an expression to its simplest form. In complex number operations, it involves combining like terms and using known properties of numbers. After applying the distributive law, reduce your expression by:

• Grouping and calculating the real parts separately.
• Handling the imaginary parts and remembering crucial properties like \(i^2 = -1\).

For example, after distributing in our problem, the real terms become \(-14 - 15\), and the imaginary terms merge into \(-11i\). Add the adjusted real result \(-15\) to \(-14\) to finalize the real part. This provides us a cleaned-up, easy-to-understand expression of \(-29 - 11i\).

Mastering simplification enhances clarity and ensures you reach the correct solution, every time.