Problem 23 - Algebra and Trigonometry

Question

Evaluate the expression and write the result in the form \(a+b i .\) $$ (3-4 i)(5-12 i) $$

Step-by-Step Solution

Answer

The expression evaluates to \(-33 - 56i\).

Step 1: Expand the Expression

Begin by distributing each term in the first complex number to each term in the second complex number. This involves using the distributive property: \((3-4i)(5-12i) = 3 \times 5 + 3 \times (-12i) + (-4i) \times 5 + (-4i) \times (-12i)\).

Step 2: Perform Multiplications

Carry out the multiplications from Step 1:- \(3 \times 5 = 15\)- \(3 \times (-12i) = -36i\)- \((-4i) \times 5 = -20i\)- \((-4i) \times (-12i) = 48i^2\)

Step 3: Simplify Using \(i^2 = -1\)

Replace \(i^2\) with \(-1\) in \(48i^2\):\(48i^2 = 48 \times (-1) = -48\).

Step 4: Combine Real and Imaginary Parts

Now, combine all the parts, remembering to combine like terms:- Real parts: \(15 - 48 = -33\)- Imaginary parts: \(-36i - 20i = -56i\)

Step 5: Write the Final Answer

The expression simplifies to the final result:\(-33 - 56i\).

Key Concepts

Distributive Property

In mathematics, the distributive property allows us to multiply a sum by distributing the multiplication over each individual term within parentheses. When dealing with complex numbers like \[\begin{pmatrix} (3-4i)(5-12i) \end{pmatrix}\]this property becomes handy by enabling us to expand expressions easily. Essentially, you apply the multiplication to each term:

• Multiply the first components: \(3 \times 5\)
• Multiply the first term of one binomial by the second term of the other: \(3 \times (-12i)\)
• Multiply the second term of one binomial by the first term of the other: \((-4i) \times 5\)
• Finally, multiply the two second terms: \((-4i) \times (-12i)\)

By breaking down the expression this way, each step becomes manageable. This systematic approach is crucial, especially when dealing with imaginary numbers, ensuring no term is forgotten.

Imaginary Unit

The imaginary unit \(i\) is a fundamental concept when working with complex numbers. It is defined as the square root of \(-1\). This means:\[ i^2 = -1 \]In the given expression, the imaginary unit appears in each term, making it crucial to handle it carefully. When multiplying two imaginary units, such as \((-4i) \times (-12i)\), we end up with \(48i^2\). Recognizing that \(i^2 = -1\) allows us to convert \(i^2\) into a real number. By doing so, the expression becomes fully solvable, and we can proceed to simplify.Handling \(i\) requires paying attention to these transformations, as they are essential in simplifying expressions and achieving the standard form \(a + b i\). Knowing this conversion from \(i^2\) to \(-1\) is a key step in mastering complex number arithmetic.

Simplifying Expressions

Simplification is the stage where everything comes together. Once the terms are multiplied, as shown through the distributive property, each resulting component, whether real or involving \(i\), needs to be combined. In our example, the expanded form:\[ 15 - 36i - 20i + 48i^2 \]gets simplified by first addressing the \(i^2\) term. Here, \(48i^2\) simplifies to \(-48\) by substituting \(i^2 = -1\). After this, focus turns to combine like terms:

• Real numbers: Combine constant numbers, such as \(15\) and \(-48\).
• Imaginary numbers: Combine coefficients of \(i\), like \(-36i\) and \(-20i\).

Finally, write the expression in the standard form \(a + bi\). For this example, the final result is \(-33 - 56i\). By following these steps, simplifying expressions becomes a structured task, allowing complex numbers to be more intuitively understood and manipulated.