Problem 27 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The result is \(-33 - 56i\).

Step 1: Distribute the Terms

Multiply the expression using the distributive property: \[(3-4i)(5-12i) = 3(5) + 3(-12i) - 4i(5) - 4i(-12i)\].

Step 2: Calculate Each Product

Perform the multiplications: \[= 15 - 36i - 20i + 48i^2\]. Recall that the product of imaginary unit \(i^2\) is \(-1\), so \(48i^2 = 48(-1) = -48\).

Step 3: Combine Like Terms

Combine the real parts and the imaginary parts separately:\[= (15 - 48) + (-36i - 20i) \].

Step 4: Simplify the Expression

Complete the calculations:\[= -33 - 56i\].

Key Concepts

Distributive Property

The distributive property is a key math concept that allows us to simplify expressions by multiplying each term inside a bracket by a term outside the bracket. In the context of complex numbers, this property helps in breaking down expressions involving binomials, such as

• (3-4i) and (5-12i)

Let's use our example:

• (3-4i)(5-12i)

With the distributive property, we'll multiply each part of the first complex number by each part of the second complex number, like this:

• 3 multiplied by 5
• 3 multiplied by -12i
• -4i multiplied by 5
• -4i multiplied by -12i

These multiplications form the expanded expression:

• 3(5) + 3(-12i) - 4i(5) - 4i(-12i)

As you can see, the distributive property breaks down the problem into manageable mini-calculations.

Imaginary Unit

In mathematics, the imaginary unit, denoted as \(i\), is a vital element in complex numbers. The imaginary unit’s defining property is:

• \(i^2 = -1\)

This property helps simplify terms involving \(i^2\). In our exercise, after applying the distributive property, one of the products involves \(i^2\):

• 48i^2

By knowing that \(i^2 = -1\), we substitute 48i² with:

• 48(-1) = -48

This conversion changes the imaginary component to a real number, simplifying our expression and bringing clarity to combining different types of terms in the subsequent steps.

Combining Like Terms

After breaking down the expression and simplifying the terms using the imaginary unit, the next step is to combine like terms. This means gathering together terms that share common features—specifically, real parts with real parts, and imaginary parts with imaginary parts.
In the expanded expression from our exercise, we have:

• Real numbers: 15 and -48
• Imaginary numbers: -36i and -20i

To combine:

• Add the real numbers: 15 - 48 = -33
• Add the imaginary numbers: -36i - 20i = -56i

Bringing these together gives the final simplified expression:

• -33 - 56i

This is how we write the result in standard complex number form, \(a + bi\), with \(-33\) as the real part and \(-56i\) as the imaginary part.