Problem 18 - Intermediate Algebra

Question

Multiply or divide. See Example 2. $$ \sqrt{-11} \cdot \sqrt{-3} $$

Step-by-Step Solution

Answer

The product is \(-\sqrt{33}\).

Step 1: Understanding the Problem

We need to multiply two square roots: \( \sqrt{-11} \) and \( \sqrt{-3} \), both of which involve negative numbers. This suggests that we will be working within the realm of complex numbers.

Step 2: Express Square Roots in Terms of Imaginary Numbers

Recall that the square root of a negative number can be expressed using the imaginary unit \( i \), where \( i^2 = -1 \). Therefore, \( \sqrt{-11} = \sqrt{11} \cdot i \) and \( \sqrt{-3} = \sqrt{3} \cdot i \).

Step 3: Multiply the Square Roots

Next, we multiply the two expressions: \( (\sqrt{11} \cdot i)(\sqrt{3} \cdot i) = \sqrt{11} \cdot \sqrt{3} \cdot i^2 = \sqrt{33} \cdot i^2 \).

Step 4: Simplify Using Imaginary Unit

Since \( i^2 = -1 \), the expression \( \sqrt{33} \cdot i^2 \) simplifies to \( \sqrt{33} \cdot (-1) = -\sqrt{33} \).

Key Concepts

Understanding the Imaginary Unit

The imaginary unit, represented by the symbol \( i \), is a fundamental concept in dealing with complex numbers. It is defined as the square root of \(-1\), which is not a real number. Instead, it's an innovative concept that allows us to extend the real number system. Here are a few key points:

• \( i^2 = -1 \), meaning when \( i \) is squared, it results in \(-1\).
• Using \( i \), we can express the square root of any negative number. For example, \( \sqrt{-4} = 2i \).
• The introduction of the imaginary unit allows us to operate within a broader numerical framework, leading to the creation of complex numbers.

Complex numbers, in turn, combine real and imaginary parts to open up new possibilities in mathematics and engineering.

Negative Square Roots and Their Expressions

In the realm of real numbers, negative square roots don't exist. However, with the imaginary unit \( i \), we can make sense of them. Let's see how:

• Consider the expression \( \sqrt{-a} \), where \( a \) is a positive real number.
• This can be rewritten using the imaginary unit as \( \sqrt{a} \cdot i \).
• For instance, \( \sqrt{-11} \) becomes \( \sqrt{11} \cdot i \), and \( \sqrt{-3} \) becomes \( \sqrt{3} \cdot i \).

By leveraging the imaginary unit, we transform previously undefined expressions into comprehensible forms. This allows us to perform calculations that include negative square roots, as shown in our exercise.

Multiplication in Complex Numbers

Multiplying complex numbers involves combining their real and imaginary components through known rules. Here's a step-by-step breakdown using our exercise:

• First, express each square root involving a negative number using the imaginary unit, resulting in \( \sqrt{-11} = \sqrt{11} \cdot i \) and \( \sqrt{-3} = \sqrt{3} \cdot i \).
• Then, multiply these expressions: \( (\sqrt{11} \cdot i)(\sqrt{3} \cdot i) = \sqrt{11} \cdot \sqrt{3} \cdot i^2 \).
• Notice that \( i \times i = i^2 = -1 \). So, the expression further simplifies to \( \sqrt{33} \cdot (-1) = -\sqrt{33} \).

Multiplication in the context of complex numbers often involves recognizing patterns with \( i \), especially \( i^2 = -1 \). This helps in simplifying results and resolving calculations that might seem cumbersome at first glance.