Problem 100 - Elementary and Intermediate Algebra

Question

Perform the operations. Write all answers in the form \(a+b i .\) $$ (3-\sqrt{-4})(4-\sqrt{-9}) $$

Step-by-Step Solution

Answer

The answer is \(6 - 17i\).

Step 1: Simplify the square roots of negative numbers

Identify and simplify the square roots of negative numbers in the expression. The expression contains \( \sqrt{-4} \) and \( \sqrt{-9} \). Recall that \( \sqrt{-1} = i \), where \( i \) is the imaginary unit.Hence, \( \sqrt{-4} = \sqrt{4} \cdot \sqrt{-1} = 2i \) and \( \sqrt{-9} = \sqrt{9} \cdot \sqrt{-1} = 3i \). So the expression becomes: \((3 - 2i)(4 - 3i)\).

Step 2: Perform the multiplication using the distributive property

Apply the distributive property (also known as the FOIL method for binomials) to multiply the two complex numbers.The expression becomes: \((3)(4) + (3)(-3i) + (-2i)(4) + (-2i)(-3i)\).

Step 3: Simplify each term

Carry out the multiplication for each term separately:1. \( 3 \times 4 = 12 \)2. \( 3 \times -3i = -9i \)3. \( -2i \times 4 = -8i \)4. \( -2i \times -3i = 6i^2 \)Remember that \( i^2 = -1 \), so \( 6i^2 = 6(-1) = -6 \).

Step 4: Combine like terms

Combine the real parts and the imaginary parts from the previous step:- Combine \( 12 \) and \( -6 \): \( 12 - 6 = 6 \).- Combine \( -9i \) and \( -8i \): \( -9i - 8i = -17i \).The expression simplifies to:\( 6 - 17i \).

Step 5: Write the final expression in standard form

Ensure the expression is in the form \( a + bi \), where \( a \) is the real part and \( b \) is the coefficient of the imaginary part \( i \). From Step 4, the expression is: \( 6 - 17i \), which is already in the required form.

Key Concepts

Imaginary Unit

In the realm of complex numbers, the imaginary unit, denoted as \( i \), is a fundamental concept used to extend the real number system. It is defined by the property: \( i^2 = -1 \). This opens up the possibility of square roots for negative numbers, which is not possible within the traditional real number system. For example, \( \sqrt{-4} \) cannot be simplified using real numbers, but by employing the imaginary unit, it becomes \( 2i \) since \( \sqrt{-4} = \sqrt{4} \times \sqrt{-1} = 2 \times i \). This allows us to express and work with numbers in the form of complex numbers, where they have both real and imaginary components.
Understanding the imaginary unit is crucial when solving equations or performing operations that involve the square roots of negative numbers. This concept plays a key role in defining complex numbers and enables a broader range of mathematical operations.

Distributive Property

The distributive property is an essential algebraic rule that allows us to multiply a single term by a collection of terms contained within parentheses. When applied to complex numbers, particularly in the context of binomials, it facilitates the process of expansion.
For example, given an expression like \((3 - 2i)(4 - 3i)\), the distributive property lets us expand it as \( (3 \cdot 4) + (3 \cdot -3i) + (-2i \cdot 4) + (-2i \cdot -3i) \).
When using the distributive property, each term from the first binomial multiplies every term from the second. This technique simplifies handling complex products and forms the basis for methods such as FOIL, offering a structured approach to operate on complex expressions.

FOIL Method

The FOIL method is a specific application of the distributive property for multiplying two binomials. "FOIL" stands for First, Outer, Inner, Last, representing the pairs of terms from each binomial that are multiplied together.
In the case of complex numbers, applying FOIL to the expression \((3 - 2i)(4 - 3i)\) involves:

• First: \( 3 \times 4 = 12 \)
• Outer: \( 3 \times -3i = -9i \)
• Inner: \(-2i \times 4 = -8i \)
• Last: \( -2i \times -3i = 6i^2 \)

The method ensures all terms are accounted for in a controlled manner.
After using FOIL, it's important to combine like terms, especially considering the role of \( i^2 = -1 \), which affects the final simplification.

Standard Form of Complex Numbers

The standard form of a complex number is expressed as \(a + bi\), where \(a\) represents the real part and \(b\) is the coefficient of the imaginary part \(i\). This format ensures clarity when communicating the components of complex solutions.
After expanding and simplifying a complex expression, as in the example \(3 - 2i\) and \(4 - 3i\), the final result \(6 - 17i\) is already in standard form.
Presenting complex numbers in standard form allows for straightforward addition, subtraction, and comparison. It provides a full view of both the real and imaginary components, and is crucial for accurately solving mathematical problems involving complex numbers.