Problem 28 - Algebra and Trigonometry

Question

Evaluate the expression and write the result in the form a bi. $$ \left(\frac{2}{3}+12 i\right)\left(\frac{1}{6}+24 i\right) $$

Step-by-Step Solution

Answer

The evaluated expression is \( \frac{-2591}{9} + 18i \).

Step 1: Identify the expression

We need to evaluate the expression \( \left(\frac{2}{3} + 12i\right) \left(\frac{1}{6} + 24i\right) \). This requires multiplying two complex numbers.

Step 2: Use the distributive property

Use the distributive property (also known as FOIL for binomials) to multiply the terms: First, \( \frac{2}{3} \times \frac{1}{6} \), Outside, \( \frac{2}{3} \times 24i \), Inside, \( 12i \times \frac{1}{6} \), and Last, \( 12i \times 24i \).

Step 3: Multiply the First terms

Calculate \( \frac{2}{3} \times \frac{1}{6} \). This gives \( \frac{2}{18}=\frac{1}{9} \).

Step 4: Multiply the Outside terms

Calculate \( \frac{2}{3} \times 24i \). This gives \( \frac{48}{3}i = 16i \).

Step 5: Multiply the Inside terms

Calculate \( 12i \times \frac{1}{6} \). This gives \( \frac{12}{6}i = 2i \).

Step 6: Multiply the Last terms

Calculate \( 12i \times 24i \). Since \( i^2 = -1 \), this gives \( 288i^2 = 288(-1) = -288 \).

Step 7: Combine all terms

Combine all the calculated terms: \( \frac{1}{9} + 16i + 2i - 288 \). The imaginary terms can be combined: \( 18i \).

Step 8: Simplify the expression

Combine the real parts: \( \frac{1}{9} - 288 \). To do this, convert \( -288 \) to a fraction, \( -288 = \frac{-2592}{9} \), then add it to \( \frac{1}{9} \) to get \( \frac{-2591}{9} \).

Step 9: Write final answer

The expression \( \left(\frac{2}{3} + 12i\right) \left(\frac{1}{6} + 24i\right) \) is evaluated as \( \frac{-2591}{9} + 18i \).

Key Concepts

Distributive Property

When multiplying complex numbers, the distributive property is a fundamental tool. This property allows you to multiply a single term by each term within parentheses. In this case, we use it to multiply two binomials containing complex numbers.
The distributive property states that for any numbers or expressions, if you have to multiply a number by a sum of numbers, it is equivalent to multiplying every addend separately and then adding the results:
\[ a(b + c) = ab + ac \]
This means each part of the first set of parentheses in our original problem is distributed across each part of the second:

• First, multiply the first terms from each parenthesis.
• Outside, multiply the outer terms from each pair of parentheses.
• Inside, multiply the inner terms from each pair of parentheses.
• Last, multiply the last terms from each set of parentheses.

FOIL Method

The FOIL method is a specific application of the distributive property for multiplying two binomials. It's an acronym that helps remember the steps needed: First, Outside, Inside, Last.
Let's expand the expression using FOIL:

• **First**: Multiply the first terms from each binomial: \( \frac{2}{3} \times \frac{1}{6} = \frac{1}{9} \).
• **Outside**: Multiply the outermost terms in each binomial: \( \frac{2}{3} \times 24i = 16i \).
• **Inside**: Multiply the innermost terms in each binomial: \( 12i \times \frac{1}{6} = 2i \).
• **Last**: Multiply the last terms from each binomial: \( 12i \times 24i = 288i^2 \). Remember, \( i^2 = -1 \), so this becomes \( -288 \).

Combine these results to complete the multiplication, and further simplify the components to reach a final answer.

Imaginary Unit

In the world of complex numbers, the imaginary unit, represented as \( i \), is quite essential.
By definition, \( i \) represents the square root of \(-1\). The core property of \( i \) is:

• \( i^2 = -1 \)

This property becomes crucial when dealing with complex numbers, especially in multiplication. For example, when multiplying \( 12i imes 24i \), we end up with \( 288i^2 \). Using the property of \( i \), this simplifies to \( 288 \times (-1) \), which is \( -288 \). Thus, understanding the imaginary unit simplifies the calculation considerably and helps solve problems involving complex numbers, making sure that you correctly combine real and imaginary parts.