Problem 44 - A Graphical Approach to College Algebra

Question

Multiply or divide as indicated. Simplify each answer. $$\frac{\sqrt{-12} \cdot \sqrt{-6}}{\sqrt{8}}$$

Step-by-Step Solution

Answer

The simplified expression is \\(-3\\).

Step 1: Simplify the Expression Involving Imaginary Numbers

Recall that \(\sqrt{-a} = i\sqrt{a}\) where \(i\) is the imaginary unit. The expression \(\sqrt{-12} \cdot \sqrt{-6}\) can be rewritten as \(i\sqrt{12} \cdot i\sqrt{6} = i^2\sqrt{12 \cdot 6} = -1\sqrt{72} = -\sqrt{72}\).

Step 2: Simplify the Radical

Now simplify the expression \(-\sqrt{72}\). We know \(72 = 36 \times 2\), so \(\sqrt{72} = \sqrt{36 \times 2} = \sqrt{36} \cdot \sqrt{2} = 6\sqrt{2}\). Hence, \(-\sqrt{72} = -6\sqrt{2}\).

Step 3: Simplify the Denominator

The denominator is \(\sqrt{8}\). Represent \(8\) as \(4 \times 2\). Thus, \(\sqrt{8} = \sqrt{4 \times 2} = \sqrt{4}\sqrt{2} = 2\sqrt{2}\).

Step 4: Divide the Simplified Numerator by the Simplified Denominator

We now have the fraction \(-\frac{6\sqrt{2}}{2\sqrt{2}}\). When we divide \(6\sqrt{2}\) by \(2\sqrt{2}\), the term \(\sqrt{2}\) cancels out in the numerator and the denominator, leaving us with \(-\frac{6}{2}\).

Step 5: Simplify the Result

Simplify \(-\frac{6}{2}\) which equals \(-3\). This is the simplified value of the original expression.

Key Concepts

Square Roots

Square roots are fundamental in math and are denoted by the radical symbol \( \sqrt{} \). When you take the square root of a number, you're finding another number that, when multiplied by itself, gives you the original number. For example, the square root of 9 is 3, because \( 3 \times 3 = 9 \).

There are two types of square roots: the principal (positive) square root and the negative square root. In mathematics, when we refer to the square root, we often mean the principal square root.

However, when dealing with negative numbers inside square roots, we use imaginary numbers. The square root of a negative number involves the imaginary unit \( i \), where \( i^2 = -1 \). For instance, \( \sqrt{-4} = 2i \), because \( i^2 \times 2^2 = -4 \).

Understanding square roots helps in simplifying and solving many mathematical expressions, especially those involving radicals and imaginary numbers.

Radicals

Radicals are expressions that include the root symbol \( \sqrt{} \). The most common radical is the square root, but you can also have cube roots, fourth roots, and so on. When working with radicals, it's important to simplify them as much as possible to make calculations and operations easier to manage.

For example, consider the expression \( \sqrt{72} \). This can be broken down into \( \sqrt{36 \times 2} \), which simplifies to \( \sqrt{36} \cdot \sqrt{2} \). Since \( \sqrt{36} = 6 \), the simplified form becomes \( 6\sqrt{2} \).

Having a good grasp of radicals can also streamline operations like adding, subtracting, or multiplying them. To multiply radicals, you multiply the numbers under the radicals, just as shown when multiplying \( \sqrt{-12} \) and \( \sqrt{-6} \).

In equations these radicals often appear, demanding correct simplification to reach a conventional result or solution.

Simplifying Expressions

Simplifying expressions is a crucial part of mathematics, making complex expressions more manageable and easier to understand. This process involves reducing an expression to its simplest form. In the case of our exercise, simplifying involves both radicals and imaginary numbers.

Let's consider how the expression \( \frac{\sqrt{-12} \cdot \sqrt{-6}}{\sqrt{8}} \) is simplified. First, convert each negative root to an imaginary number using \( \sqrt{-a} = i\sqrt{a} \), then combine and simplify the radicals. In our example, this involves \( i^2 \sqrt{12 \cdot 6} = -\sqrt{72} \).

After simplifying the numerator and denominator separately, we simplify the fraction further by canceling out common factors. In the original problem, \( \sqrt{2} \) was canceled from both the numerator and denominator.

Finally, it's vital to ensure all operations adhere to mathematical laws, leading to a cleaner and more comprehensible result. In our case, the result of simplifying was \(-3\), which is the expression in its simplest form.