Problem 35 - College Algebra with Corequisite Support

Question

For the following exercises, perform the indicated operation and express the result as a simplified complex number. $$ \sqrt{-9}+3 \sqrt{-16} $$

Step-by-Step Solution

Answer

The simplified complex number is \(15i\).

Step 1: Simplify Square Roots of Negative Numbers

To simplify the square roots of negative numbers, we first express them in terms of the imaginary unit, \(i\), where \(i = \sqrt{-1}\). Thus, \(\sqrt{-9} = \sqrt{9} \times \sqrt{-1} = 3i\) and \(\sqrt{-16} = \sqrt{16} \times \sqrt{-1} = 4i\).

Step 2: Substitute Simplified Values

Substitute the simplified expressions from Step 1 into the original equation: \(\sqrt{-9} + 3 \sqrt{-16} = 3i + 3 \times 4i\).

Step 3: Perform Multiplication

Multiply \(3\) and \(4i\) to get \(3 \times 4i = 12i\). Now the expression becomes \(3i + 12i\).

Step 4: Combine Like Terms

Combine the like terms \(3i\) and \(12i\) to get \(15i\). Thus, the expression simplifies to \(15i\).

Key Concepts

Imaginary Unit

The imaginary unit, often denoted as \( i \), is a fundamental concept in complex numbers. It helps us comfortably work with the square roots of negative numbers, which do not have real solutions. By definition, \( i \) is equal to \( \sqrt{-1} \). This may seem strange at first, as it challenges the rule that the square of any real number is positive.

The main power of using \( i \) in mathematics lies in its ability to transform negative square roots into a form we can manipulate. For any negative number \(-a\), its square root can be expressed as \( \sqrt{a} \times i \). This transformation allows us to express more complex equations and solve problems that involve negative square roots with ease.

Here are some basic properties of \( i \):

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

Understanding \( i \) and its powers will help simplify many operations and problems involving complex numbers.

Simplifying Square Roots

Simplifying square roots, especially those of negative numbers, is a process that enables manipulation and understanding of complex problems. When faced with \( \sqrt{-9} \) and \( \sqrt{-16} \), these numbers need to be expressed through the use of the imaginary unit \( i \) to make them easier to work with.

Let's break it down:

• For \( \sqrt{-9} \), separate it into \( \sqrt{9} \times \sqrt{-1} \). We know \( \sqrt{9} = 3 \), and \( \sqrt{-1} = i \), so \( \sqrt{-9} = 3i \).
• Similarly, for \( \sqrt{-16} \), express it as \( \sqrt{16} \times \sqrt{-1} \). Since \( \sqrt{16} = 4 \), we have \( \sqrt{-16} = 4i \).

This method of multiplying the square root of the positive part of the number by \( i \) simplifies further calculations and operations involving negative square roots.

Combining Like Terms

Combining like terms is a key step in simplifying expressions, especially when dealing with complex numbers. To identify like terms, look for terms that have the same variables raised to the same power. In the context of complex numbers, like terms often involve the imaginary unit \( i \).

Consider the expression \( 3i + 12i \). Both terms involve the imaginary unit \( i \) and are therefore 'like terms.' This means they can be added or subtracted together in a straightforward manner just like regular algebraic terms.
Here's what to do:

• Add the coefficients of \( i \), which are 3 and 12 in this example.
• The result is \( 3 + 12 = 15 \), so \( 3i + 12i \) simplifies to \( 15i \).

By combining like terms, complex expressions can often be simplified greatly, making them much easier to read and understand.