Problem 36 - College Algebra

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 simplified complex number is \(-22i\).

Step 1: Simplify the square root of negative numbers

Recall that \( \sqrt{-1} = i \). Therefore, \( \sqrt{-4} = \sqrt{-1 \cdot 4} = \sqrt{4} \cdot \sqrt{-1} = 2i \), and \( \sqrt{-25} = \sqrt{-1 \cdot 25} = \sqrt{25} \cdot \sqrt{-1} = 5i \).

Step 2: Substitute the simplified forms

Substitute \( 2i \) for \( \sqrt{-4} \) and \( 5i \) for \( \sqrt{-25} \) into the original expression:\[ -\sqrt{-4} - 4 \sqrt{-25} \rightarrow -(2i) - 4(5i). \]

Step 3: Distribute and simplify

Apply the multiplication operations:\[ -(2i) = -2i, \]and\[ 4(5i) = 20i. \]

Step 4: Combine like terms

Combine the imaginary terms:\[ -2i - 20i = -22i. \]

Step 5: Express as a simplified complex number

The expression \( -22i \) is already in the form of a simplified complex number, where the real part is 0.

Key Concepts

Simplifying Square Roots

When dealing with square roots, especially those of negative numbers, it's essential to understand how they behave. Normally, the square root of a positive number is straightforward. However, when it comes to negative numbers, things get a little bit tricky. This is where the concept of the imaginary unit, represented as \( i \), becomes crucial.

• For any positive number \( a \), the square root is expressed as \( \sqrt{a} \).
• However, for negative numbers like \( -a \), the square root is expressed as \( \sqrt{-a} = \sqrt{a} \cdot i \).

This representation allows us to simplify expressions that involve square roots of negative numbers by extracting the \( i \) from the radical sign, which signifies the imaginary unit.

Imaginary Unit

The imaginary unit, denoted by \( i \), is defined as the square root of \(-1\). It provides a way to extend the real number system to solve equations that do not have solutions in real numbers, like \( x^2 + 1 = 0 \).

• \( i^2 = -1 \)
• \( i^3 = (-1) \cdot i = -i \)
• \( i^4 = 1 \)

These properties of \( i \) are cyclical, meaning they repeat every four powers. The imaginary unit is pivotal in forming complex numbers, which are written as \( a + bi \), where \( a \) and \( b \) are real numbers, and \( bi \) is the product of \( b \) with the imaginary unit.

Operations with Complex Numbers

Working with complex numbers involves performing operations similar to those done with real numbers, but with an added consideration for the imaginary part. In our exercise, the goal was to simplify and combine terms to reach a final simplified form.Key operations include:

• Addition and Subtraction: Combine like terms by adding or subtracting the real parts and the imaginary parts separately, like in combining \(-2i - 20i = -22i\).
• Multiplication: Distribute terms as you would in polynomial multiplication, keeping track of the imaginary unit. Remember, \( i^2 = -1 \), which often simplifies the expression.

In this exercise, the real part ended up being zero, and the resulting complex number was purely imaginary \(-22i\). Understanding these operations helps in simplifying expressions and solving equations involving complex numbers.