Problem 23 - Precalculus Student Solutions Manual 5th

Question

Write each number as the product of a real number and i. $$-\sqrt{-18}$$

Step-by-Step Solution

Answer

-3\sqrt{2}i

Step 1: Simplify the expression inside the square root

Rewrite the expression as \(-\sqrt{-18} = -\sqrt{18}\sqrt{-1}\). Remember that \(\sqrt{-1} = i\).

Step 2: Simplify the square root of the positive number

Simplify \(\sqrt{18}\) which can be written as \(\sqrt{9 \cdot 2} = \sqrt{9} \cdot \sqrt{2} = 3\sqrt{2}\).

Step 3: Combine the results

Combine the results from Step 1 and Step 2 to get \(-\sqrt{18} \cdot \sqrt{-1} = -(3\sqrt{2}) \cdot i = -3\sqrt{2}i\).

Key Concepts

Understanding the Imaginary Unit

When working with complex numbers, the imaginary unit, denoted as \(i\), is essential. The imaginary unit is defined as \(i = \sqrt{-1}\). This simple yet powerful concept allows us to deal with the square roots of negative numbers, which are not possible with real numbers alone.
The existence of \(i\) expands our number system from real numbers to complex numbers. Complex numbers are in the form \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit.

Square Roots and Their Properties

Square roots allow us to find a number which, when multiplied by itself, gives the original number. For instance, \( \sqrt{9} = 3\) because \(3 \times 3 = 9\). When we deal with negative numbers under a square root, we use \(i\), the imaginary unit. For example, \( \sqrt{-4} = \sqrt{4} \cdot \sqrt{-1} = 2i\).
Breaking down square roots can simplify complex problems. Let's consider \( \sqrt{18}\). We can factor 18 into its square roots: \( 18 = 9 \cdot 2 \). This allows us to write \(\sqrt{18} = \sqrt{9} \cdot \sqrt{2} = 3 \sqrt{2}\). This step-by-step simplification makes handling expressions easier.

Simplifying Expressions with Complex Numbers

Simplifying complex number expressions often involves breaking them down into easier parts before reassembling them. Let's revisit the provided exercise: convert \( -\sqrt{-18} \) into a product of a real number and \( i \).
First, recognize that \( \sqrt{-18} = \sqrt{18} \cdot \sqrt{-1} \). We know \( \sqrt{-1} = i \), transforming the expression into \( \sqrt{18} i \).
Next, simplify \( \sqrt{18} \). Factoring it as \( 18 = 9 \cdot 2 \), we get \( \sqrt{18} = 3 \sqrt{2} \). Therefore, substituting back, we obtain \( -3 \sqrt{2} i \).
By breaking down the problem into manageable steps, we can simplify complex expressions effectively. Remember to combine like terms and simplify square roots first before reintroducing the imaginary component.