Problem 42 - College Algebra

Question

In Exercises \(29-44,\) perform the indicated operations and write the result in standard form. $$\sqrt{-12}(\sqrt{-4}-\sqrt{2})$$

Step-by-Step Solution

Answer

The expression \(\sqrt{-12}(\sqrt{-4}-\sqrt{2})\) simplifies to \(-4\sqrt{3} - 2i\sqrt{6}\) in standard form.

Step 1: Transform the square roots

We must remember that the square root of a negative number can be expressed as multiples of \(i\). We simplify \(\sqrt{-12}\), \(\sqrt{-4}\), and \(\sqrt{2}\) individually. \(\sqrt{-12} = i\sqrt{12} = 2i\sqrt{3}\), \(\sqrt{-4} = 2i\) and \(\sqrt{2}\) remains the same as it is positive.

Step 2: Perform Operations

Subtract \(\sqrt{2}\) from \(2i\). \(2i - \sqrt{2}\). Now we multiply this result by \(2i\sqrt{3}\). We end with \(2i\sqrt{3}(2i - \sqrt{2})\)

Step 3: Distribute and Simplify

\(2i\sqrt{3}(2i - \sqrt{2})\) simplifies to \(4i^2\sqrt{3} - 2i\sqrt{6}\). \(i^2\) is equal to \(-1\), so it changes the sign to negative for the first part of the expression: \(4i^2\sqrt{3} = -4\sqrt{3}\). Finally, the expression in standard form turns into \(-4\sqrt{3} - 2i\sqrt{6}\).

Key Concepts

Imaginary Numbers

Imaginary numbers are a fascinating concept in mathematics. They help us perform operations with numbers that are not part of the real number system. At the core of this concept is the imaginary unit, denoted as \( i \), which is defined as the square root of \( -1 \). This means \( i^2 = -1 \).

When we encounter a negative number under a square root, like \( \sqrt{-12} \), we use \( i \) to express it. For instance, \( \sqrt{-12} \) is rewritten as \( i\sqrt{12} \). Imaginary numbers allow us to extend the real number system into what we call 'complex numbers', which consist of a real part and an imaginary part.

They are crucial when solving algebraic equations that don't have real solutions. This expansion of the number system is essential in advanced mathematics, engineering, and physics. Imaginary numbers provide the key to understanding many complex phenomena.

Square Roots

Square roots are a fundamental concept in mathematics. They ask the question: "What number, when multiplied by itself, gives the original number?". For positive numbers, the square root is straightforward. For example, \( \sqrt{4} = 2 \) because \( 2 \times 2 = 4 \).

However, the concept gets interesting with negative numbers. Normally, no real number squared gives a negative, which is where imaginary numbers come in handy. We encounter expressions like \( \sqrt{-4} \), which we transform into \( 2i \) by using the imaginary unit \( i \). The multiplication of \( i \) helps us work with these otherwise undefined operations.

This method reveals that negative numbers have 'imaginary square roots', leading us to the realm of complex numbers, where operations like addition, subtraction, multiplication, and even division of roots are possible. Understanding how to handle square roots is vital for simplifying and solving algebraic problems involving radical expressions.

Standard Form Transformation

In algebra, standard form typically refers to writing numbers in a simplified and consistent way. For complex numbers, this involves expressing them as \( a + bi \), where \( a \) is the real part and \( b \) is the imaginary part. Converting expressions into standard form allows for easy manipulation and comparison.

When transforming expressions, operations such as distribution, combining like terms, and simplifying using \( i^2 = -1 \) are applied. For example, while solving \( \sqrt{-12}(\sqrt{-4} - \sqrt{2}) \), we distribute and simplify to find \(-4\sqrt{3} - 2i\sqrt{6}\). This simplifies further by recognizing that processing \( i^2 \) results in a negative real number, changing the sign, thus aiding in restructuring into the standard form.

Mastering this transformation is important for solving equations that require a specific format for easier interpretation and is useful in both pure mathematics and its applications in real-world scenarios.