Problem 16 - College Algebra with Modeling and Visualization

Question

Simplify by using the imaginary unit \(i\). $$ \sqrt{-8} \cdot \sqrt{-8} $$

Step-by-Step Solution

Answer

The expression simplifies to \(-8\).

Step 1: Understanding the Imaginary Unit

The imaginary unit is defined as \( i \), where \( i^2 = -1 \). This means that \( \sqrt{-1} = i \). Any square root of a negative number can be rewritten using \( i \).

Step 2: Simplifying Each Square Root

Simplify \( \sqrt{-8} \) using the imaginary unit. Note that \( -8 = -1 \times 8 \), so \( \sqrt{-8} = \sqrt{-1 \times 8} = \sqrt{-1} \times \sqrt{8} = i \times \sqrt{8} \).

Step 3: Combine the Expressions

Now that \( \sqrt{-8} = i\sqrt{8} \), we replace both instances of \( \sqrt{-8} \) in the original expression: \( \sqrt{-8} \cdot \sqrt{-8} = (i\sqrt{8}) \cdot (i\sqrt{8}) \).

Step 4: Multiply the Expressions

Multiply \( (i\sqrt{8}) \cdot (i\sqrt{8}) \). First, multiply the imaginary units: \( i \cdot i = i^2 \). Then multiply the square roots: \( \sqrt{8} \cdot \sqrt{8} = 8 \). So, \( (i\sqrt{8}) \cdot (i\sqrt{8}) = i^2 \times 8 \).

Step 5: Substitute and Simplify the Imaginary Unit

Since \( i^2 = -1 \), substitute and simplify: \( i^2 \times 8 = -1 \times 8 = -8 \).

Key Concepts

Understanding Square Roots

Square roots are mathematical operations that give a number which, when multiplied by itself, results in the original number. For example, the square root of 9 is 3, because 3 multiplied by 3 results in 9. However, things get interesting when dealing with negative numbers. Normally, we cannot take the square root of a negative number because no real number, when squared, gives a negative result.
This is where the imaginary unit, denoted as \(i\), becomes useful. The imaginary unit \(i\) is defined such that \(i^2 = -1\). With this definition, the square root of a negative number can be expressed in terms of \(i\). For example:

• \(\sqrt{-1} = i\)
• \(\sqrt{-9} = \sqrt{-1 \times 9} = \sqrt{-1} \times \sqrt{9} = i \times 3 = 3i\)

This transformation allows us to work with the square roots of negative numbers by treating them in the realm of complex numbers.

Diving into Complex Numbers

Complex numbers are an extension of the real numbers that include the imaginary unit \(i\). A complex number is expressed in the form \(a + bi\), where \(a\) and \(b\) are real numbers. Here, \(a\) is known as the real part, and \(bi\) represents the imaginary part.
The beauty of complex numbers lies in their ability to solve equations that have no real solutions. For instance, the equation \(x^2 + 1 = 0\) has no real solutions, as no real number squared is negative, but it has complex solutions: \(x = i\) and \(x = -i\).
Through the use of complex numbers, we can perform operations on negative square roots smoothly. For our exercise, \(\sqrt{-8}\) can be expressed as \(i\sqrt{8}\), thus utilizing both the imaginary unit and the real number to represent complex values.

Simplifying Algebraic Expressions

Algebra simplification is the process of rewriting expressions in a simpler or more compact form. In our exercise, simplification uses properties of square roots and imaginary numbers to achieve a more straightforward result.
We started with an expression, \(\sqrt{-8} \cdot \sqrt{-8}\). First, we expressed each \(\sqrt{-8}\) as \(i\sqrt{8}\), then applied the rules of multiplication for complex numbers:

• Multiply the imaginary parts: \(i \cdot i = i^2 = -1\).
• Multiply the square roots: \(\sqrt{8} \cdot \sqrt{8} = 8\).

Combining these results, we find \(i^2 \times 8 = -1 \times 8 = -8\). By recognizing and applying these algebraic rules step-by-step, any potentially confusing expression involving complex numbers can be simplified with ease.