Problem 15 - College Algebra with Modeling and Visualization

Question

Simplify by using the imaginary unit \(i\). $$ \sqrt{-5} \cdot \sqrt{-5} $$

Step-by-Step Solution

Answer

The product is \( -5 \).

Step 1: Identify the Use of Imaginary Unit

The imaginary unit is defined as \( i = \sqrt{-1} \). This allows us to simplify square roots of negative numbers. In this problem, both instances of \( \sqrt{-5} \) can be rewritten using the imaginary unit.

Step 2: Rewrite Each Square Root Using the Imaginary Unit

We rewrite \( \sqrt{-5} \) as \( \sqrt{5} \cdot i \). This is because \( \sqrt{-5} = \sqrt{5} \cdot \sqrt{-1} = \sqrt{5} \cdot i \).

Step 3: Multiply the Two Expressions

To find \( \sqrt{-5} \cdot \sqrt{-5} \), multiply the two rewritten square roots: \( (\sqrt{5} \cdot i) \cdot (\sqrt{5} \cdot i) = \sqrt{5} \cdot \sqrt{5} \cdot i \cdot i \).

Step 4: Simplify the Multiplication

Simplify the expression by multiplying: \( \sqrt{5} \cdot \sqrt{5} = 5 \), and since \( i \cdot i = i^2 \), and \( i^2 = -1 \), the entire expression becomes \( 5 \cdot (-1) = -5 \).

Key Concepts

Complex Numbers

Complex numbers are numbers that have both a real and an imaginary part. The imaginary part includes the imaginary unit, denoted as \( i \), where \( i = \sqrt{-1} \). This concept is crucial because it allows mathematicians to deal with square roots of negative numbers, which are not possible with traditional real numbers.

Complex numbers are often represented in the form \( a + bi \), where \( a \) is the real part, and \( bi \) is the imaginary part. In the context of this exercise, when you encounter \( \sqrt{-5} \), you're really looking at a complex number since it involves the imaginary unit.

Working with complex numbers is similar to working with two-dimensional vectors:

• The real part defines one dimension.
• The imaginary part defines the other dimension.

This unique structure allows us to perform operations like addition, subtraction, and multiplication on complex numbers just like we do with regular numbers.

Square Roots

Square roots represent a value that, when multiplied by itself, gives the original number. For example, the square root of 25 is 5 because \( 5 \cdot 5 = 25 \). However, when dealing with negative numbers under a square root, things are not straightforward since you can't multiply two identical real numbers to get a negative number.

This is where the imaginary unit \( i \) comes in handy. The square root of a negative number, such as \( \sqrt{-5} \), can be rewritten with \( i \). Specifically, \( \sqrt{-5} = \sqrt{5} \cdot i \). Breaking it down into these pieces allows us to perform calculations that involve complex numbers.

Thus, when you have square roots of negative numbers, always consider rewriting them using \( i \), turning potentially confusing expressions into manageable forms.

Multiplication of Radicals

Multiplication of radicals involves combining square roots through multiplication, which can sometimes simplify the expression or make it possible to deal with complex numbers. In this exercise, \( \sqrt{-5} \cdot \sqrt{-5} \) is such a case.

By rewriting each \( \sqrt{-5} \) as \( \sqrt{5} \cdot i \), you can multiply them using the properties of radicals and imaginary numbers:

• Combine the radicals: \( \sqrt{5} \cdot \sqrt{5} = 5 \).
• Multiply the imaginary units: \( i \cdot i = i^2 \), and since \( i^2 = -1 \), this portion of the product becomes \(-1\).

Finally, put it all together to get the result \( 5 \cdot (-1) = -5 \).

Thus, multiplication of radicals using imaginary numbers not only simplifies expressions but also shows the powerful flexibility of complex numbers in mathematical operations.