Problem 36 - A Graphical Approach to Precalculus with Limits

Question

Multiply or divide as indicated. Simplify each answer. $$\sqrt{-5} \cdot \sqrt{-15}$$

Step-by-Step Solution

Answer

The product is \\(-5\sqrt{3}\\).

Step 1: Understand the Expression

We have \(\sqrt{-5} \times \sqrt{-15}\). These are square roots of negative numbers, which leads us to use imaginary numbers.

Step 2: Express in Terms of Imaginary Numbers

Recall that \(\sqrt{-1} = i\). Thus, \(\sqrt{-5} = \sqrt{5}\cdot i\) and \(\sqrt{-15} = \sqrt{15}\cdot i\).

Step 3: Multiply the Imaginary Parts

Using the expressions from Step 2, multiply them together: \(\sqrt{-5} \times \sqrt{-15} = (\sqrt{5} \cdot i) \times (\sqrt{15} \cdot i)\).

Step 4: Simplify the Product

First, multiply the square roots: \(\sqrt{5} \times \sqrt{15} = \sqrt{75}\). Then, multiply the imaginary units: \(i \times i = i^2\), which equals \(-1\) since \(i^2 = -1\).

Step 5: Solve and Simplify

Combine conclusions from Step 4: \(\sqrt{-5} \times \sqrt{-15} = \sqrt{75} \times (-1)\). Simplify \(\sqrt{75} = \sqrt{25 \times 3} = 5\sqrt{3}\). Therefore, the product becomes \(-5\sqrt{3}\).

Key Concepts

Imaginary Numbers

Imaginary numbers may sound daunting at first, but they're simply a part of mathematics dealing with square roots of negative numbers. More specifically, an imaginary number is defined as the square root of a negative number. The basic unit of imaginary numbers is represented as the letter 'i', where

• \( i = \sqrt{-1} \)

This means that when you see \( \sqrt{-5} \), you can express it using imaginary numbers as \( \sqrt{5} \cdot i \). By understanding this concept, you can approach problems involving square roots of negative numbers more confidently. Complex numbers combine real and imaginary numbers, and are written in the form \( a + bi \), where \( a \) is real and \( bi \) represents the imaginary part.

Square Roots

Square roots are another important concept in this exercise. A square root of a number \( x \) is a value that, when multiplied by itself, gives \( x \). For example, \( \sqrt{9} = 3 \) because \( 3 \times 3 = 9 \).
However, when dealing with negative numbers, we need imaginary numbers to define the square root, since no real number squared equals a negative number. Thus, the square root of any negative number converts into an imaginary number.

• For instance, \( \sqrt{-15} \) becomes \( \sqrt{15} \cdot i \).

Thus, understanding square roots involves recognizing when to apply imaginary numbers, especially when the values are negative.

Multiplication

When multiplying complex expressions such as \( \sqrt{-5} \times \sqrt{-15} \), it requires careful organization and understanding of both real and imaginary parts. Start by separating out the square roots from the imaginary units:

• \( \sqrt{-5} \times \sqrt{-15} = (\sqrt{5} \cdot i) \times (\sqrt{15} \cdot i) \)

The multiplication is then performed in two steps:
1. Multiply the square root components:

• \( \sqrt{5} \times \sqrt{15} = \sqrt{75} \)

2. Multiply the imaginary components:

• \( i \times i = i^2 \), and by definition, \( i^2 = -1 \)

This results in the expression \( \sqrt{75} \times (-1) \). Understanding these steps makes multiplication of imaginary number expressions straightforward.

Simplifying Expressions

Once you have multiplied the components of the expression, simplifying is the final step. Simplification makes the expression easy to interpret and use. In our exercise, we found \( \sqrt{75} \times (-1) \) and need to simplify \( \sqrt{75} \).

• Factor \( 75 \) to recognize perfect squares: \( 75 = 25 \times 3 \).
• Thus, \( \sqrt{75} = \sqrt{25 \times 3} = 5\sqrt{3} \).

This gives us the expression \( -5\sqrt{3} \). This simplification involves breaking down numbers into their prime factors to easily identify and simplify perfect squares, which is a useful strategy for handling and simplifying square roots efficiently.