Problem 35 - A Graphical Approach to Precalculus with Limits

Question

Multiply or divide as indicated. Simplify each answer. $$\sqrt{-3} \cdot \sqrt{-8}$$

Step-by-Step Solution

Answer

The simplified expression is \(-2\sqrt{6}\).

Step 1: Identify the Components

We are given two square roots with negative numbers: \(\sqrt{-3}\) and \(\sqrt{-8}\). Both involve imaginary numbers because the square root of a negative number includes \(i\), where \(i = \sqrt{-1}\).

Step 2: Express Each Square Root with Imaginary Units

Express \(\sqrt{-3}\) and \(\sqrt{-8}\) using imaginary numbers: \(\sqrt{-3} = \sqrt{3} \cdot i\) and \(\sqrt{-8} = \sqrt{8} \cdot i\).

Step 3: Multiply the Imaginary Components

Multiply the two expressions: \(\sqrt{3} \cdot i \cdot \sqrt{8} \cdot i = \sqrt{3 \cdot 8} \cdot i^2 = \sqrt{24} \cdot i^2\).

Step 4: Simplify the Product with Imaginary Numbers

Since \(i^2 = -1\), replace \(i^2\) in the expression: \(\sqrt{24} \cdot i^2 = \sqrt{24} \cdot (-1) = -\sqrt{24}\).

Step 5: Simplify the Square Root

Simplify \(\sqrt{24}\) by breaking it into prime factors: \(\sqrt{24} = \sqrt{4 \cdot 6} = \sqrt{4} \cdot \sqrt{6} = 2\sqrt{6}\).

Step 6: Final Simplification

Combine the results from the previous steps: \(-\sqrt{24} = -2\sqrt{6}\). This is the simplified form of the expression.

Key Concepts

Imaginary Numbers

Imaginary numbers are numbers that arise when dealing with the square roots of negative numbers. In mathematics, these numbers are essential because they allow us to work with equations that have no real-number solutions. The imaginary unit is represented by the symbol \(i\), which is defined as \(i = \sqrt{-1}\). This simple concept unlocks an entirely new number system known as complex numbers.

Remember:

• Imaginary numbers are multiples of \(i\).
• Imaginary numbers, like real numbers, can be added, subtracted, and multiplied just like any other numbers.
• Combining imaginary numbers with real numbers creates complex numbers, expressed as \(a + bi\), where \(a\) and \(b\) are real numbers.

This understanding helps handle the square roots of negative numbers, such as in the original exercise.

Square Roots

Square roots are operations that find a number which, when multiplied by itself, gives the original number. For example, the square root of 9 is 3, because \(3 \times 3 = 9\). When taking the square root of negative numbers, we enter the domain of complex numbers, traditionally represented using the imaginary unit \(i\).

Here's how to handle it:

• For positive numbers, it's straightforward: \(\sqrt{9} = 3\).
• For negative numbers, introduce \(i\): \(\sqrt{-9} = \sqrt{9} \cdot i = 3i\).

In the exercise, we had \(\sqrt{-3} \cdot \sqrt{-8}\), which translates into complex numbers \(\sqrt{3}i\) and \(\sqrt{8}i\).
This concept is key to solving equations involving negative square roots, allowing us to express and simplify complex solutions.

Simplification

Simplification is all about making expressions as straightforward as possible. When dealing with complex numbers, this means breaking them down into their simplest form.

Let's break it down:

• Convert the expression into components you understand (e.g., \(\sqrt{3} \cdot \sqrt{8} \cdot i^2\)).
• Simplify the square roots into prime factors (e.g., \(\sqrt{24} = \sqrt{4 \cdot 6}\)).
• Use known values, such as \(i^2 = -1\), to simplify the expression further.
• Combine and simplify where possible to reach the final, simplest form.

The final result in our exercise was \(-2\sqrt{6}\). The process of simplification helps consolidate complex expressions into manageable terms for clearer understanding and calculation.

Iota

Iota is simply another name for the imaginary unit \(i\), which forms the foundation for imaginary and complex numbers. Understanding \(iota\) is crucial for dissecting expressions involving the square root of negative numbers.

Essentials of \(i\):

• \(i = \sqrt{-1}\), a fundamental principle that defines imaginary numbers.
• \(i^2 = -1\) is a key feature used in simplification processes, as seen in our exercise.
• \(i\) can combine with real numbers to create complex numbers, used broadly in various fields of math and engineering.

The role of \(i\) or iota in mathematics extends beyond basic operations and provides solutions where real numbers alone aren't enough. Familiarity with \(iota\) makes navigating complex numbers more intuitive and practical, especially in expressions like \(\sqrt{-3} \cdot \sqrt{-8}\).