Problem 36 - College Algebra with Corequisite Support

Question

For the following exercises, perform the indicated operation and express the result as a simplified complex number. $$ -\sqrt{-4}-4 \sqrt{-25} $$

Step-by-Step Solution

Answer

The result is \(-22i\).

Step 1: Simplify Square Roots of Negative Numbers

The square root of a negative number involves the imaginary unit \( i \), where \( i = \sqrt{-1} \). Therefore, convert \( \sqrt{-4} \) to an imaginary number: \( \sqrt{-4} = \sqrt{-1 \times 4} = \sqrt{-1} \times \sqrt{4} = 2i \). Similarly, \( \sqrt{-25} = \sqrt{-1 \times 25} = \sqrt{-1} \times \sqrt{25}= 5i \).

Step 2: Substitute and Perform Indicated Operations

After substituting the imaginary forms from Step 1, the expression is: \[ -\sqrt{-4} - 4 \sqrt{-25} = -(2i) - 4(5i) \].

Step 3: Simplify Expression

Now simplify the expression: \[ -(2i) - 20i = -2i - 20i = -22i \]. This is the simplified complex number.

Key Concepts

Imaginary Unit

The imaginary unit, denoted as \( i \), is a mathematical concept that allows us to work with the square roots of negative numbers. This might seem peculiar at first, since there is no real number whose square is negative.

The defining property of the imaginary unit is:

• \( i = \sqrt{-1} \)
• \( i^2 = -1 \)

This means that when you multiply the imaginary unit by itself, the result is -1. Introducing \( i \) expands our number system beyond real numbers, making it possible to express and calculate the square roots of negative numbers.
In more advanced mathematics, \( i \) becomes a critical component of complex numbers, which combine real numbers and imaginary numbers.

Square Roots of Negative Numbers

Working with square roots of negative numbers initially challenges our understanding of numbers. In the realm of real numbers, taking the square root of a negative number is impossible. Here is where the concept of the imaginary unit \( i \) comes into play.

To find the square root of a negative number like \( -4 \):

• Rewrite \( \sqrt{-4} \) as \( \sqrt{-1 \times 4} \).
• Apply the property \( \sqrt{a \times b} = \sqrt{a} \cdot \sqrt{b} \).
• Recognize \( \sqrt{-1} \) as \( i \).

So, \( \sqrt{-4} = \sqrt{-1} \cdot \sqrt{4} = 2i \). Similarly, \( \sqrt{-25} = 5i \).
Understanding this process is key to working with complex numbers, which allow calculations beyond the capabilities of the real number system.

Simplifying Expressions with Imaginary Numbers

Simplifying expressions with imaginary numbers involves similar steps to those used with real numbers, but always considering the unique property of \( i \).

Let's look at an expression like \( -\sqrt{-4} - 4\sqrt{-25} \):

• First, convert each square root into its simplifies imaginary form: \( \sqrt{-4} = 2i \) and \( \sqrt{-25} = 5i \).
• Substitute these values back into the expression: \( -(2i) - 4(5i) \).
• Perform basic arithmetic operations, keeping in mind that \( i \) acts as a regular variable here: \( -(2i) - 20i = -2i - 20i = -22i \).

The final result is a simplified complex number expressed in terms of \( i \). Remember, even though it appears different, simplification rules for imaginary expressions remain consistent with those for algebraic expressions concerning real numbers.