Problem 6 - Introductory Algebra for College Students

Question

Express each number in terms of i. $$\sqrt{-12}$$

Step-by-Step Solution

Answer

The solution is \(2i\sqrt{3}\).

Step 1: Express Negative Root in terms of 'i'

We can write \(\sqrt{-12}\) as \(\sqrt{-1 * 12}\). The square root of -1 is \(i\) so it becomes \(i\sqrt{12}\).

Step 2: Simplify Root

\(\sqrt{12}\) can be simplified to \(2\sqrt{3}\) so it simplifies to \(i * 2\sqrt{3}\).

Key Concepts

Imaginary Unit

When dealing with complex numbers, the imaginary unit, denoted as \(i\), plays a central role. It is defined as the square root of -1.

This definition is crucial because it allows us to extend the concept of square roots to negative numbers, which we otherwise cannot do with real numbers alone.

For example:

• \(i^2 = -1\), which follows directly from its definition.
• Because \(i = \sqrt{-1}\), any time you encounter the square root of a negative number, it generally involves \(i\).

Understanding \(i\) as an imaginary unit helps us bridge the gap between real numbers and complex numbers, paving the way for solving a wider array of mathematical problems.

Simplifying Radicals

Simplifying radicals is an important skill when working with both real and complex numbers. The goal is to express the radical in its simplest form.

To simplify \( \sqrt{12} \), we first look for the largest perfect square that divides evenly into the number under the square root.

Steps for simplifying \( \sqrt{12} \):

• Identify the factors of 12: 1, 2, 3, 4, 6, 12.
• The largest perfect square is 4, and \(4 \times 3 = 12\).

Now, rewrite \( \sqrt{12} \) as \( \sqrt{4 \times 3} \). Since \(\sqrt{4} = 2\), you can simplify \( \sqrt{12} \) to \(2\sqrt{3}\).

Simplifying radicals helps make expressions easier to work with, especially when combined with imaginary numbers.

Square Roots of Negative Numbers

Square roots of negative numbers can seem challenging at first, but with the help of the imaginary unit \(i\), they become manageable.

When you encounter a square root of a negative number, the key is to first isolate the negative part by using \(i\).

For example:

• Consider \( \sqrt{-12} \).
• Start by rewriting it as \( \sqrt{-1 \cdot 12} \).
• We know that \( \sqrt{-1} = i \), so replace \(\sqrt{-1}\) with \(i\), giving you \(i\sqrt{12}\).
• Finally, simplify \(\sqrt{12}\) to \(2\sqrt{3}\) to get \(i \cdot 2\sqrt{3}\).

By using \(i\), you effectively bring the negative component outside of the square root, allowing you to handle it just like any other radical expression.