Problem 21 - Intermediate Algebra

Question

Express each number in terms of \(i\). $$ \sqrt{-24} $$

Step-by-Step Solution

Answer

The expression \\(\sqrt{-24}\\) in terms of \\(i\\) is \\(2i\sqrt{6}\\).

Step 1: Recognize the Structure

Understand that \(-24\) can be rewritten using the imaginary unit \(i\), which is defined as \(i = \sqrt{-1}\). This will help convert the expression \(\sqrt{-24}\) into terms involving \(i\).

Step 2: Factor Negative One Out

Rewrite \(-24\) as \((-1)\times24\). This allows us to separate the negative part of the square root, which will enable us to incorporate \(i\).

Step 3: Apply the Definition of Imaginary Unit

Use the definition of \(i\) to rewrite \(\sqrt{-1}\) as \(i\). Hence, \(\sqrt{-1} = i\), and the expression becomes \(\sqrt{(-1) \times 24} = \sqrt{-1} \cdot \sqrt{24}\).

Step 4: Simplify the Square Root

Calculate the square root of 24. Factor 24 into \(2 imes 2 imes 2 imes 3\), recognizing that you can take out pairs of numbers to simplify: \(\sqrt{4 imes 6} = \sqrt{4} \cdot \sqrt{6} = 2\sqrt{6}\).

Step 5: Combine the Results

Multiply \(i\) by the simplified square root, resulting in the final expression. The square root of \(-24\) is thus \(i \cdot 2 \sqrt{6}\) or simply \(2i \sqrt{6}\).

Key Concepts

Imaginary Unit

When it comes to complex numbers, the imaginary unit is a concept you cannot ignore. The imaginary unit, denoted as \(i\), is defined as \(i = \sqrt{-1}\). This is a crucial idea because it allows us to handle square roots of negative numbers, which are not possible under the set of real numbers alone.

• The main role of \(i\) is to extend the real number system to include solutions to equations that would otherwise have no real-number solutions.
• Any imaginary number can be expressed as \(bi\) where \(b\) is a real number.

A typical example is \(\sqrt{-24}\). By using \(i\), you can express this as \(i\sqrt{24}\), which provides a way to work with negative roots in practical applications like engineering or physics.Remember, the imaginary unit is imaginary in name only; it plays a real and integral role in the realm of mathematics.

Square Root

Understanding square roots is fundamental to grasping the essence of complex numbers. The square root of a number \(x\) is a value that, when multiplied by itself, yields \(x\). For positive numbers, this is straightforward. However, it becomes challenging for negative numbers.

• For example, \(\sqrt{4}\) is \(2\) because \(2 \times 2 = 4\).
• With negative numbers, you employ the imaginary unit: \(\sqrt{-24} = i\sqrt{24}\) since \(\sqrt{-1} = i\).

The concept extends beautifully into extracting square roots from composite numbers, such as \(\sqrt{24}\), which can be simplified further by identifying the prime factors and grouping them: \(\sqrt{4 \times 6} = 2\sqrt{6}\). This process helps avoid blind calculations, offering a simpler form as evidenced in many mathematical solutions.

Simplifying Radicals

Simplifying radicals is an invaluable skill that aids in making complex expressions manageable. Simplification involves expressing the radical in its simplest form, so it contains no perfect square factors other than 1.The idea is straightforward. When simplifying \(\sqrt{24}\), you start by looking for factors that are perfect squares:

• For instance, 24 can be factored as \(2 \times 2 \times 6\) or \(4 \times 6\).
• Recognizing \(4\) as a perfect square, you can simplify \(\sqrt{24}\) to \(2\sqrt{6}\).

By simplifying \(\sqrt{24}\) before multiplying by \(i\) in the original problem, the expression \(i \cdot 2\sqrt{6}\) or simply \(2i\sqrt{6}\) emerges as a far more elegant and usable form. This showcases the beauty and utility in simplifying radicals, setting the stage for more complex algebraic work.