Problem 9 - Algebra 2 and Trigonometry

Question

In \(3-18,\) write each number in terms of \(i\) $$ \sqrt{-12} $$

Step-by-Step Solution

Answer

\(\sqrt{-12} = 2i\sqrt{3}\).

Step 1: Understand Imaginary Numbers

First, we need to recall that the imaginary unit, denoted as \(i\), is defined by \(i^2 = -1\). This means any negative number under a square root can be expressed in terms of \(i\).

Step 2: Simplify the Square Root

The expression \(\sqrt{-12}\) can be rewritten by factoring out \(-1\) from the square root, giving us \(\sqrt{-1 \times 12}\). This expression can be split into two separate square roots: \(\sqrt{-1} \times \sqrt{12}\).

Step 3: Solve for \(\sqrt{-1}\)

We know that \(\sqrt{-1} = i\). So, the first part of our product becomes \(i\).

Step 4: Simplify \(\sqrt{12}\)

The square root of 12 can be simplified. We break it down into its factors: \(12 = 4 \times 3\). Hence, \(\sqrt{12} = \sqrt{4 \times 3} = \sqrt{4} \times \sqrt{3} = 2 \cdot \sqrt{3}\).

Step 5: Combine Results

Multiply \(i\) by the simplified \(\sqrt{12}\). Thus, \(\sqrt{-12} = i \cdot 2\sqrt{3}\).

Step 6: Conclusion: Write the Answer in Standard Form

Combine the results to express the number in terms of \(i\): \(\sqrt{-12} = 2i\sqrt{3}\).

Key Concepts

Square Roots

A square root helps us find a number that, when multiplied by itself, gives the original number. For example, the square root of 9 is 3, because 3 multiplied by 3 equals 9. However, finding the square root of negative numbers involves a special concept. Normally, we can't take the square root of a negative number within the realm of real numbers because no real number squared will give a negative result.
Enter the world of imaginary numbers. We introduce the imaginary unit "i" as the square root of -1. This allows us to express square roots of negative numbers. For instance, \[\sqrt{-4} = \sqrt{4 \times -1} = \sqrt{4} \times \sqrt{-1} = 2i.\]Understanding how to manipulate these roots underpins working with complex numbers.

Complex Numbers

Complex numbers come into play when we deal with solutions that involve imaginary numbers. A complex number consists of two parts: a real part and an imaginary part. The general form of a complex number is\[a + bi,\]where:

• \(a\) is the real part
• \(bi\) is the imaginary part

For example, in the calculation of \(\sqrt{-12}\), which simplifies to \[2i\sqrt{3},\]\(0\) can be thought of as the real part, and \(2 \sqrt{3}\) as the coefficient on \(i\).Complex numbers are essential in advanced mathematics, such as signal processing and control theory, where they help model oscillations and waves.

Algebraic Expressions

Algebraic expressions contain numbers, variables, and operations. Simplifying these expressions is a core skill in algebra. Consider the expression derived from the exercise:\[\sqrt{-12} = 2i\sqrt{3}.\]Here, we have an imaginary component \(2i\) and a real multiplication outside the square root \(\sqrt{3}\). Simplifying such expressions often involves factoring numbers, substituting known values, and performing arithmetic operations. With imaginary numbers, knowing that \(i^2 = -1\) is fundamental in simplifying or manipulating expressions.Working comfortably with algebraic expressions is critical for solving equations and understanding relationships between variables in mathematics. This skill extends to many fields such as engineering, physics, and computer science, where expressing quantities accurately is key.