Problem 8 - College Algebra with Modeling and Visualization

Question

Simplify by using the imaginary unit \(i\). $$ \sqrt{-32} $$

Step-by-Step Solution

Answer

The simplified form is \(4i \sqrt{2}\).

Step 1: Identify the Imaginary Unit

The imaginary unit is represented by \( i \). It is defined by \( i = \sqrt{-1} \). Any negative number inside the square root can be expressed in terms of \( i \).

Step 2: Separate the Negative and Positive Parts

Given \( \sqrt{-32} \), we can separate it into \( \sqrt{-1} \times \sqrt{32} \). This makes it easier to simplify because \( \sqrt{-1} \) is equal to \( i \).

Step 3: Simplify the Positive Part

Now, focus on \( \sqrt{32} \). Notice that 32 can be factored into 16 and 2, where \( 16 \) is a perfect square. Thus, \( \sqrt{32} = \sqrt{16 \times 2} = \sqrt{16} \times \sqrt{2} = 4 \times \sqrt{2} \).

Step 4: Combine Results

Combine the parts from Step 2 and Step 3: \( \sqrt{-1} \times \sqrt{32} = i \times 4 \times \sqrt{2} = 4i \sqrt{2} \).

Key Concepts

Complex Numbers

Complex numbers form an essential part of mathematics, extending the idea of numbers beyond the real line. A complex number is expressed in the form \(a + bi\), where:

• \(a\) is the real part
• \(b\) is the imaginary part
• \(i\) is the imaginary unit

The imaginary unit \(i\) is defined as \(i = \sqrt{-1}\). This allows mathematicians to work with square roots of negative numbers. For example, to find the square root of \(-32\) as shown in our exercise, we rewrite it in terms of \(i\), leading us into the world of complex numbers. This process of involving the imaginary unit turns a nonsensical problem for real numbers into a manageable one.

Square Roots

Square roots are a fundamental mathematical operation where we find a number that, when multiplied by itself, gives the original number. However, when dealing with negative numbers under the square root sign, we enter the realm of imaginary numbers.
In our exercise, we are tasked with simplifying \(\sqrt{-32}\). Traditionally, square roots of negative numbers are not defined in the set of real numbers, but with the help of the imaginary unit \(i\), we manage square roots of negatives beautifully.

• We express \(\sqrt{-32}\) as \(\sqrt{-1} \times \sqrt{32}\)
• Here, \(\sqrt{-1}\) is given by \(i\)
• The square root of the positive portion, \(\sqrt{32}\), needs further simplification

Step-by-step, using the property \(i^2 = -1\), we comfortably handle negative roots, showing the utility of imaginary numbers in algebra.

Algebraic Simplification

Algebraic simplification involves breaking down expressions to their most straightforward form, making them easier to work with and understand.
In the context of our original problem, we are simplifying \(\sqrt{-32}\), which involves:

• Recognizing and separating the square root of a negative number into parts
• Simplifying these parts individually
• Combining them back to produce the simplest version of the original expression

Breaking down \(\sqrt{32}\) into \(\sqrt{16 \times 2}\) helps utilize the known perfect square \(16\), as \(\sqrt{16} = 4\). So, \(\sqrt{32} = 4 \times \sqrt{2}\).
This step-by-step simplification transforms the expression into \(4i \sqrt{2}\), demonstrating how seemingly complex problems can be tackled methodically, offering insight into both the elegance and utility of algebra.