Problem 43 - College Algebra

Question

In Exercises \(29-44,\) perform the indicated operations and write the result in standard form. $$(3 \sqrt{-5})(-4 \sqrt{-12})$$

Step-by-Step Solution

Answer

The answer to the multiplication operation is \( 12\sqrt{60} \).

Step 1: Identify Imaginary Numbers

First, we have to recognize that the square root of a negative number results in an imaginary number. If we write \( \sqrt{-1} \) as \( i \), then the numbers in the problem are expressed as \( 3i\sqrt{5} \) and \( -4i\sqrt{12} \).

Step 2: Multiply Numbers

Multiply the numbers as usual, realizing that \( i^2 \) equals -1: \( (3i\sqrt{5})(-4i\sqrt{12}) = -12i^2\sqrt{60}. \)

Step 3: Convert to Standard Form

Because \( i^2 = -1 \), we can simplify further: \( -12i^2\sqrt{60} = 12\sqrt{60}. \)

Key Concepts

Imaginary Numbers

Imaginary numbers are fascinating mathematical concepts that help us solve problems involving negative square roots. When you encounter an expression like \( \sqrt{-1} \), it might seem impossible at first. However, by defining this as \( i \), which stands for an imaginary unit, we unlock new ways of dealing with these tricky numbers. The imaginary unit \( i \) is specifically defined such that \( i^2 = -1 \). This definition allows us to work with negative square roots just like regular numbers.
Consider when you find \( \sqrt{-5} \) or \( \sqrt{-12} \) in your exercises. To write them in terms of imaginary numbers, they become \( i\sqrt{5} \) and \( i\sqrt{12} \), respectively. Imaginary numbers, combined with real numbers, form complex numbers, which are written in the form \( a + bi \). Here, \( a \) is the real part and \( bi \) is the imaginary part.

Multiplication of Complex Numbers

Multiplying complex numbers involves extending the distributive property and carefully managing the imaginary units. When you multiply complex numbers, such as \( (3i\sqrt{5})(-4i\sqrt{12}) \), you treat the imaginary parts with special care. Here’s how it works:

• First, multiply the regular numbers: \( 3 \times -4 = -12 \).
• Next, multiply the imaginary units: \( i \times i = i^2 \). Remember, \( i^2 = -1 \).
• Finally, multiply the square roots: \( \sqrt{5} \times \sqrt{12} = \sqrt{60} \).

When you bring all these parts together, the expression becomes \( -12i^2\sqrt{60} \).
The crucial part here is recognizing \( i^2 = -1 \), which transforms \( -12i^2 \) into \( 12 \). Therefore, \( -12i^2\sqrt{60} \) simplifies into a real number component with \( 12\sqrt{60} \).

Standard Form of Complex Numbers

The standard form of complex numbers is essential for clarity and further complex calculations. It is expressed as \( a + bi \), where \( a \) and \( b \) are real numbers and \( i \) is the imaginary unit. This form ensures uniformity and ease of interpretation for complex numbers.
After performing operations with complex numbers, such as multiplication, we often need to convert the result into standard form. In the exercise we discussed, the initial simplification \( -12i^2\sqrt{60} \) converts to \( 12\sqrt{60} \) due to the properties of \( i^2 \). However, if there were non-zero imaginary parts after simplification, they would be expressed in the form \( a + bi \).
To ensure the result is in standard form, simplify all parts and explicitly express both the real and imaginary components. This makes the complex number easy to read and ready for use in further mathematical contexts.