Problem 16 - Intermediate Algebra

Question

Simplify. See Example 1. $$ \sqrt{-63} $$

Step-by-Step Solution

Answer

The simplified form of \(\sqrt{-63}\) is \(3i\sqrt{7}\).

Step 1: Express the Number Under the Square Root as a Product

We begin by factoring the number under the square root, 63. Notice that 63 can be factored into 63 = 9 * 7. Since \(-63 = -1 \times 63\), we express this as \(\sqrt{-63} = \sqrt{-1 \times 9 \times 7}\).

Step 2: Separate the Square Root Across the Product

The property \(\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}\) allows us to separate the square root of a product into the product of square roots.Therefore, \(\sqrt{-1 \times 9 \times 7}\) can be rewritten as \(\sqrt{-1} \cdot \sqrt{9} \cdot \sqrt{7}\).

Step 3: Simplify Each Part of the Product

The square root of -1 is denoted as \(i\), the imaginary unit. The square root of 9 is 3, because \(3^2 = 9\). Thus, we have \(\sqrt{-1} = i\), \(\sqrt{9} = 3\), and \(\sqrt{7}\) remains as it is.

Step 4: Multiply the Results

Combine the simplified square roots: \(i \cdot 3 \cdot \sqrt{7}\). This gives us \(3i\sqrt{7}\) as the simplified form of \(\sqrt{-63}\).

Key Concepts

Square Roots

The symbol \( \sqrt{} \) indicates a square root. It's used to find the number that, when multiplied by itself, will give the original number under the square root.
For instance, \( \sqrt{9} = 3 \) because \( 3^2 = 9 \). Similarly, each positive number has a non-negative square root. However, what if we need to find the square root of a negative number like \(-63\)?
This leads us to complex numbers and the value of \( i \), the imaginary unit defined as \( \sqrt{-1} = i \).
With this definition, we can express square roots of negative numbers in terms of \( i \), making it possible to deal with negative square roots in mathematical operations.

Complex Numbers

Complex numbers are numbers that have both a real part and an imaginary part. They are generally written in the form \( a + bi \), where \( a \) is the real part, and \( bi \) is the imaginary part.
The imaginary unit, \( i \), is defined by the property \( i^2 = -1 \). This allows us to work with numbers that include the square root of negative numbers.
For example, when simplifying \( \sqrt{-63} \), we can separate the components into real and imaginary parts using \( i \).

• Real part: considered as usual real numbers, like \( 3 \).
• Imaginary part: includes \( i \) like \( 3i \).

The result of simplifying \( \sqrt{-63} \) becomes \( 3i\sqrt{7} \), which blends real and imaginary parts as follows: \( 3i \) is the imaginary component, while \( \sqrt{7} \) stays a real number.

Simplifying Expressions

To simplify mathematical expressions, we reduce them to their simplest form, eliminating any unnecessary complexity.
This often involves using properties and rules, such as distributing square roots over multiplication: \( \sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b} \).
For \( \sqrt{-63} \), we factor \( -63 \) into \( -1 \times 63 \), then proceed to \( 9 \times 7 \). By applying the property of square roots and recognizing \( \sqrt{-1} = i \):

• First, separate each factor into its own square root: \( \sqrt{-1 \times 9 \times 7} = \sqrt{-1} \times \sqrt{9} \times \sqrt{7} \).
• Then simplify each unary square root: \( \sqrt{-1} = i \), \( \sqrt{9} = 3 \), and finally \( \sqrt{7} \) remains.

Combine the results to form the cleanest version: multiplying all parts gives us \( 3i\sqrt{7} \), the simplified form of the expression.