Problem 33 - Algebra for College Students

Question

Write each of the following in terms of \(i\) and simplify. For example, $$ \sqrt{-20}=i \sqrt{20}=i \sqrt{4} \sqrt{5}=2 i \sqrt{5} $$ $$\sqrt{-18}$$

Step-by-Step Solution

Answer

\(\sqrt{-18} = 3i\sqrt{2}\).

Step 1: Identify the Negative Under the Square Root

We begin by recognizing that any negative number under the square root indicates the presence of the imaginary unit \(i\), where \(i = \sqrt{-1}\). Thus, we can rewrite \(\sqrt{-18}\) as \(\sqrt{-1 \times 18}\).

Step 2: Separate the Negative

Using the property \(\sqrt{ab} = \sqrt{a} \times \sqrt{b}\), we separate the negative from the positive component: \(\sqrt{-1 \times 18} = \sqrt{-1} \times \sqrt{18}\). Simplifying \(\sqrt{-1}\) gives us \(i\), so this becomes \(i \times \sqrt{18}\).

Step 3: Simplify the Square Root

Now we focus on simplifying \(\sqrt{18}\). We find the prime factorization of 18, which is \(18 = 2 \times 3^2\). Thus, \(\sqrt{18} = \sqrt{2} \times \sqrt{3^2} = \sqrt{2} \times 3 = 3\sqrt{2}\).

Step 4: Combine the Terms

Combine \(i\) with our simplified square root result. Therefore, \(i \times \sqrt{18} = i \times 3\sqrt{2} = 3i\sqrt{2}\).

Key Concepts

Imaginary Unit

Complex numbers are an extension of the real numbers, and they introduce the famous imaginary unit, denoted as \(i\). This unit is defined by the equation \(i = \sqrt{-1}\). Understanding \(i\) is crucial because it allows us to handle square roots of negative numbers, which don't exist in the world of real numbers. When you see a negative under a square root, you know \(i\) will play a part. For instance, the expression \(\sqrt{-9}\) becomes \(i\sqrt{9}\) or \(3i\).

• \(i^2 = -1\): Multiplying \(i\) by itself results in \(-1\).
• \(i^3 = -i\): This is because \(i^3 = i^2 \times i = -1 \times i = -i\).
• \(i^4 = 1\): Multiplying \(i\) four times returns to the start: \(i^4 = (i^2)^2 = (-1)^2 = 1\).

This cyclical pattern is important. Recognizing these powers lets you handle more complex expressions involving \(i\) with ease.

Square Root Simplification

Simplifying square roots is essential for making expressions more manageable. The property \(\sqrt{ab} = \sqrt{a} \times \sqrt{b}\) is a key tool. It lets us break down complex square roots into simpler components. Consider \(\sqrt{18}\). Through simplification, we factor 18 into its prime factors \(2 \times 3^2\), allowing us to rewrite it as \(\sqrt{2} \times \sqrt{3^2}\). In turn, \(\sqrt{3^2} = 3\), simplifying the expression to \(3\sqrt{2}\).Using this approach can make calculations less intimidating and more intuitive. Remember:

• Simplify within the square root to find perfect squares.
• If the expression can be broken down, it often reveals easier numbers to work with.

Prime Factorization

Prime factorization involves breaking down a number into its basic building blocks, known as prime numbers. A prime number is one that can only be divided by itself and 1. For example, the number 18 can be broken down using prime factorization: it is \(2 \times 3 \times 3\) or \(2 \times 3^2\).Understanding prime factorization helps in simplifying square roots and solving various mathematical problems:

• Identify the prime components of any number.
• Break it down into a product of prime numbers.
• Use these factors to simplify calculations, such as roots and fractions.

Knowing how to factor numbers into primes is foundational in mathematics. It's a skill that can help you navigate algebraic simplifications and pave the way for solving more complex equations. By handling the components of a number, you can more easily simplify expressions and uncover elegant solutions.