Problem 53 - College Algebra with Modeling and Visualization

Question

Evaluate the expression with a calculator. $$ (-12.6-5.7 i)(5.1-9.3 i) $$

Step-by-Step Solution

Answer

The expression evaluates to \(-11.25 + 88.11i\).

Step 1: Understand the Expression

The given expression is in the form \((a + bi)(c + di)\), where \(a = -12.6\), \(b = -5.7\), \(c = 5.1\), and \(d = -9.3\). Our goal is to calculate \((-12.6 - 5.7i)(5.1 - 9.3i)\) using a calculator.

Step 2: Apply the Distributive Property

Apply the distributive property, also known as the FOIL (First, Outer, Inner, Last) method, to multiply the complex numbers. This involves calculating:- \( ac = (-12.6)(5.1) \)- \( ad = (-12.6)(-9.3) \)- \( bc = (-5.7)(5.1) \)- \( bd = (-5.7)(-9.3) \)

Step 3: Simplify the Real Part

Calculate the real part of the product by adding \( ac\) and \(- bd\) (because \( i^2 = -1\)):- \((-12.6)(5.1) = -64.26\)- \(-(-5.7)(-9.3) = -(-53.01) = 53.01\)Therefore, the real part is \(-64.26 + 53.01 = -11.25\).

Step 4: Simplify the Imaginary Part

Calculate the imaginary part by adding \( ad\) and \( bc\):- \((-12.6)(-9.3) = 117.18\)- \((-5.7)(5.1) = -29.07\)Thus, the imaginary part is \( 117.18 - 29.07 = 88.11 \).

Step 5: Combine Real and Imaginary Parts

Combine the real and imaginary parts from the previous steps to form the final result. The expression simplifies to:\[ -11.25 + 88.11i \]

Key Concepts

Algebraic Expressions

Algebraic expressions are a combination of numbers, variables, and mathematical operations. In the context of complex numbers, an algebraic expression can include real numbers and imaginary numbers. For example, in the expression \((-12.6 - 5.7i)(5.1 - 9.3i)\), both terms are complex numbers. They consist of a real part and an imaginary part, represented respectively by the terms without \(i\) and with \(i\). The process of evaluating this expression involves working with each part separately but in combination with each other.
Understanding how to deal with both real and imaginary parts within algebraic expressions is crucial. When manipulating these expressions, it’s important to recognize how the operations applied to real numbers also affect the imaginary components. Complex arithmetic often requires systematic methods such as the distributive property to ensure all terms are handled correctly. By mastering algebraic expressions, one can easily deal with more complex calculations involving both real and imaginary numbers.

Distributive Property

The distributive property is one of the most essential concepts in algebra. It allows us to multiply a single term across terms inside parentheses. In the case of complex numbers, the distributive property is used in the FOIL method, which stands for First, Outer, Inner, Last. This method ensures that every part of each complex number is multiplied by every part of the other.
For the expression \((-12.6 - 5.7i)(5.1 - 9.3i)\), we break it down into parts:

• First: \((-12.6)(5.1)\)
• Outer: \((-12.6)(-9.3)\)
• Inner: \((-5.7)(5.1)\)
• Last: \((-5.7)(-9.3)\)

The distributive property ensures that all parts of the first bracket interact with all parts of the second. This reduces the complexity of multiplying larger expressions. It is especially useful when dealing with complex numbers because it helps in organizing and verifying the multiple individual calculations required. By learning how to correctly apply the distributive property, students can effortlessly simplify even the most complex expressions.

Imaginary Numbers

Imaginary numbers are defined as multiples of the imaginary unit \(i\), which is the square root of -1. They may sound abstract, but they simplify computations by providing solutions to equations that don't have real roots. In the context of complex numbers, the imaginary part of a complex number is denoted by the coefficient of \(i\).
In the expression \((-12.6 - 5.7i)(5.1 - 9.3i)\), the terms \(-5.7i\) and \(-9.3i\) are the imaginary components. When multiplied, they play a crucial role in determining the final value of the expression.

• Remember: \(i^2 = -1\).
• Thus, when calculating terms like \(bd = (-5.7)(-9.3)\), the product affects the real part.

Imaginary numbers allow for the representation and manipulation of functions that cross into the complex plane, expanding the scope of analysis in many mathematical fields. By getting comfortable with imaginary numbers, students can tackle a wider array of mathematical problems, including those beyond just arithmetic with real numbers.