Problem 18 - Algebra 2 and Trigonometry

Question

In \(3-18,\) write each number in terms of \(i\) $$ -3+2 \sqrt{-9} $$

Step-by-Step Solution

Answer

The expression is \(-3 + 6i\).

Step 1: Simplify the square root of a negative number

The expression involves the square root of a negative number, \( \sqrt{-9} \). We use the imaginary unit \( i \), where \( i^2 = -1 \), to simplify it: \[ \sqrt{-9} = \sqrt{9} \cdot \sqrt{-1} = 3i \].

Step 2: Apply the simplification to the expression

Substitute \(\sqrt{-9}\) with \(3i\) in the original expression to rewrite it in terms of \(i\). The expression becomes: \[ -3 + 2 \times 3i \].

Step 3: Simplify by performing multiplication

Multiply 2 by \(3i\) to simplify: \[ 2 \times 3i = 6i \].

Step 4: Reconstruct the final expression

Now replace the original expression with the simplified terms: \[ -3 + 6i \]. This is the final form in terms of \(i\).

Key Concepts

Imaginary Unit

The imaginary unit is a fundamental concept in complex numbers that helps us handle negative square roots. In mathematics, the imaginary unit is denoted by the letter \( i \). It is defined as the square root of -1. This definition allows us to work with numbers that are not real, meaning they're not on the usual number line.To remind you:

• \( i^2 = -1 \)
• \( i^3 = -i \) (since \( i^3 = i^2 \times i = -1 \times i = -i \))
• \( i^4 = 1 \) (since \( i^4 = i^2 \times i^2 = (-1) \times (-1) = 1 \))

These properties are crucial when you're working with complex numbers, as they simplify the process of manipulating expressions that include square roots of negative numbers.

Square Root of Negative Numbers

Calculating the square root of negative numbers can seem tricky at first because, within real numbers, you can't have a square root of a negative number. This is where the imaginary unit \( i \) comes into play. With it, we can extend our number system and handle calculations that involve negative numbers under a square root.To find the square root of a negative number:

• Firstly, express the negative number as its positive counterpart multiplied by \(-1\).
• For example, for \(\sqrt{-9}\), you represent it as \(\sqrt{9 \times -1}\).
• Utilize the properties of square roots to split this into \(\sqrt{9} \cdot \sqrt{-1}\).
• Replace \(\sqrt{-1}\) with \(i\), resulting in \(3i\) for \(\sqrt{-9}\).

By understanding this process, you can handle any square root of negative numbers and express them using \(i\).

Simplification of Expressions

Simplifying complex expressions involves a few systematic steps to ensure clarity and accuracy. When you have an expression that includes both real numbers and terms involving \(i\), you want to simplify each part.Here’s how you can approach it:

• First, deal with any multiplication within the expression. For instance, in the expression \(-3 + 2 \times 3i\), start by addressing \(2 \times 3i\).
• Execute the multiplication, getting \(6i\) in the example above.
• Once all the multiplications are complete, combine terms where possible. However, since real numbers and imaginary terms are distinct, you’ll usually write them as a sum, like \(-3 + 6i\).

Remember, the goal is to present your expression clearly by performing basic arithmetic operations and retaining any remaining parts in their simplified form.