Problem 8 - Algebra 2 and Trigonometry

Question

In \(3-18,\) write each number in terms of \(i\) $$ \sqrt{-8} $$

Step-by-Step Solution

Answer

The expression \( \sqrt{-8} \) in terms of \(i\) is \(2i\sqrt{2}\).

Step 1: Recognize the Negative Under the Square Root

The expression \( \sqrt{-8} \) involves taking the square root of a negative number. Recall that when you have a negative number under a square root, you can factor out \(-1\) and use the imaginary unit \(i\) where \(i^2 = -1\). Therefore, we can express the square root of a negative number as the product of \(i\) and the square root of the positive value.

Step 2: Separate the Square Root into Imaginary and Real Components

Write \( \sqrt{-8} \) as \( \sqrt{-1} \times \sqrt{8} \). Recall that \( \sqrt{-1} = i \), so this becomes \(i \times \sqrt{8} \).

Step 3: Simplify the Square Root

To simplify \( \sqrt{8} \), recognize that 8 can be written as \(4 \times 2\). This allows you to write \( \sqrt{8} = \sqrt{4 \times 2} \). This can further be broken down as \( \sqrt{4} \times \sqrt{2} \), where \( \sqrt{4} = 2\). Thus, \( \sqrt{8} = 2\sqrt{2} \).

Step 4: Combine Terms

Now substitute back the simplified form of \( \sqrt{8} \) into the expression \(i \times \sqrt{8}\). Therefore, \( \sqrt{-8} = i \times 2\sqrt{2} = 2i\sqrt{2} \).

Key Concepts

Imaginary Unit

Understanding the imaginary unit, represented as \(i\), is crucial when working with complex numbers. The imaginary unit is defined by the property \(i^2 = -1\).
This definition helps to deal with square roots of negative numbers. Normally, taking a square root of a negative number isn't possible in the set of real numbers. However, using \(i\), it's easy to express these values.
For example, the square root of \(-1\) is simply \(i\). This becomes the building block for expressing any square root of negative numbers, like \(\sqrt{-8}\), in terms of \(i\).
In essence, \(i\) serves as a way to navigate the seemingly impossible world of negative roots.

Simplifying Square Roots

Simplifying square roots is about finding an equivalent, often more recognizable form of a square root. For instance, let's consider \(\sqrt{8}\): it can be written in a simpler format.
The goal is to break down numbers inside a square root into smaller factors to simplify the expression. Here's how this is done:

• First, factor the number inside the square root into its prime factors or recognizable squares, like \(4 \times 2\).
• Then, apply the square root to each factor: \(\sqrt{4} \times \sqrt{2}\).
• Since \(\sqrt{4} = 2\), you arrive at \(2\sqrt{2}\).

This simplified form, \(2\sqrt{2}\), makes calculations easier and expressions more straightforward.

Factoring Negative Numbers

When dealing with negative numbers under a square root, the trick is to factor out the negative sign using the imaginary unit \(i\).
This is because a negative inside a square root means the number doesn't have a real square root. Here’s how you handle it:

• First, express the square root of the negative number as the product of \(\sqrt{-1}\) and the square root of the corresponding positive number. For example, \(\sqrt{-8}\) becomes \(\sqrt{-1} \times \sqrt{8}\).
• Then use the definition \(\sqrt{-1} = i\), allowing you to rewrite the expression with \(i\).
• This turns \(\sqrt{-8}\) into \(i \times \sqrt{8}\) or \(2i\sqrt{2}\) when simplified.

This method ensures a smooth transition from negative to positive factors, enabling easier calculations within the realm of complex numbers.