Problem 46 - College Algebra Essentials

Question

In Exercises \(45-50,\) perform the indicated operation(s) and write the result in standard form. $$ (8+9 i)(2-i)-(1-i)(1+i) $$

Step-by-Step Solution

Answer

Therefore, the solution in the standard form for the given expression \( (8+9 i)(2-i)-(1-i)(1+i) \) is \(25 + 10i\).

Step 1: Distribute the Multiplication

The first step is to distribute in both the expressions: \( (8 + 9i)(2 - i) \) equates to \(16 - 8i + 18i - 9i^2\),and \( (1 - i)(1 + i) \) equates to \(1 + i - i - i^2\).

Step 2: Simplify Expressions

Simplify both expressions:For \(16 - 8i + 18i - 9i^2\), combine like terms and change \(i^2\) to \(-1\) (as \(i^2 = -1\)), obtaining: \(16 + 10i + 9 = 25 + 10i\).For \(1 + i - i - i^2\), again combine like terms and change \(i^2\) to \(-1\), obtaining: \(1 - 1 = 0\).

Step 3: Subtract the Simplified Expressions

Now subtract \(25 + 10i\) and \(0\) as per the original expression, giving us the solution \(25 + 10i - 0 = 25 + 10i\).

Key Concepts

Distributive Property

The distributive property is a foundational algebra concept used to simplify expressions where $$ one term is multiplied by two or more terms inside parentheses. It states:

• For real numbers, if you have an expression like \(a(b + c)\), it becomes \(ab + ac\).

In the world of complex numbers, the same rule applies, even when dealing with imaginary units. Consider the expression \((8 + 9i)(2 - i)\). Using the distributive property, we multiply each term in the first complex number by each term in the second:

• \(8 \cdot 2 = 16\)
• \(8 \cdot (-i) = -8i\)
• \(9i \cdot 2 = 18i\)
• \(9i \cdot (-i) = -9i^2\)

Reaching this stage, the complex multiplication is split into manageable pieces, setting up the next step in simplification.

Simplifying Expressions

Simplifying expressions, especially those with complex numbers, involves combining like terms and $ $ substituting known values.

• First, recognize that $i^2 = -1$ for all calculations.

Let's simplify $16 - 8i + 18i - 9i^2$. Combine like terms:

• $-8i + 18i = 10i$

Substitute $-9i^2$ with $9$ (since $-9(-1) = 9$):

• This transforms the expression to $16 + 10i + 9$.

The expression simplifies to $25 + 10i$ by combining $16 + 9$. For the expression $(1 - i)(1 + i)$, simplify similarly:

• $1 - i + i - i^2$ becomes $1 - (-1) = 2$.

Combining these results streamlines the process to reach the answer efficiently.

Imaginary Unit

The imaginary unit \(i\) is a mathematical concept used to extend real numbers into the complex plane. It$$ is defined by the property \(i^2 = -1\). Here's why it's crucial:

• Imaginary numbers arise naturally when solving equations like \(x^2 + 1 = 0\).
• The solution \(x = \pm i\) expands mathematics beyond the reals.

In terms of calculations, the imaginary unit allows expressions such as \(-9i^2\) in this exercise to be simplified quickly to \(9\).

• This substitution transforms complex arithmetic into something more straightforward by leveragingknown values.

Understanding how \(i\) works is key to mastering many complex number problems, allowing you to focus on technique rather than the mechanics of calculation.