Problem 21 - Algebra 2 and Trigonometry

Question

In \(19-34,\) write each sum or difference in terms of \(i\) $$ \sqrt{-144}+\sqrt{-1} $$

Step-by-Step Solution

Answer

The sum is \(13i\).

Step 1: Identify the components

Recognize that both terms under the square roots are negative numbers, which means we'll need to use the imaginary unit \(i\). Recall that \(i\) is the imaginary unit where \(i^2 = -1\).

Step 2: Simplify \(\sqrt{-144}\)

Rewrite \(\sqrt{-144}\) as \(\sqrt{144 \cdot (-1)}\). This can be simplified by separating the square root: \(\sqrt{144} \cdot \sqrt{-1}\). Since \(\sqrt{144} = 12\) and \(\sqrt{-1} = i\), we have \(\sqrt{-144} = 12i\).

Step 3: Simplify \(\sqrt{-1}\)

The square root of \(-1\) is simply \(i\) by definition. Therefore, \(\sqrt{-1} = i\).

Step 4: Add the expressions

Combine the expressions from Steps 2 and 3: \(12i + i\). This can be simplified further to \(13i\).

Key Concepts

Imaginary Unit

In mathematics, the imaginary unit, often denoted by the letter \(i\), is a cornerstone concept when dealing with complex numbers. It allows us to extend the real numbers to include square roots of negative numbers, which are not otherwise definable within the context of real numbers.
\(i\) is defined by the property that \(i^2 = -1\). This unique property is what makes operations with \(i\) possible and meaningful. The use of \(i\) transforms numbers that were once thought impossible to compute, such as the square root of a negative number, into something that we can work with mathematically.
Researchers and engineers use the imaginary unit in various fields, including electrical engineering and physics, where complex numbers are applied in analyzing waveforms and alternating currents. \(i\) plays an essential role, similar to its significance in purely mathematical endeavors.

Square Roots of Negative Numbers

Typically, when we take the square root of a negative number, we encounter a problem in the realm of real numbers. This is because no real number squared gives a negative result. The key to dealing with this problem is the introduction of the imaginary unit \(i\).
When taking the square root of a negative number like \(-144\), we separate it into \(-1 \times 144\). This allows us to utilize the imaginary unit.
Here's how it's done:

• Separate the expression under the square root: \( \sqrt{-144} = \sqrt{144 \cdot (-1)} \).
• Apply the property of square roots to separate these factors: \( \sqrt{144} \cdot \sqrt{-1} \).
• Recognize that \( \sqrt{144} = 12 \) and \( \sqrt{-1} = i \), which simplifies our expression to \( 12i \).

Thus, the square root of a negative number gets simplified into an expression involving \(i\), making computations and subsequent algebraic procedures straightforward.

Simplifying Complex Expressions

Combining terms involving the imaginary unit \(i\) demands careful arithmetic following the rules of algebra. When tasked with adding or subtracting complex numbers, one should simply combine like terms, often involving \(i\).
If we consider the expression \( \sqrt{-144} + \sqrt{-1} \) from the given exercise, we first rewrite these components:

• \(\sqrt{-144} = 12i\) as determined by our earlier steps.
• \(\sqrt{-1} = i\).

Our task then turns to simplifying \( 12i + i \). These are like terms because they both involve \(i\).
We can add them simply by combining their coefficients:\[ 12i + i = (12 + 1)i = 13i \]Consolidating complex expressions helps in minimizing any potential errors and provides clarity in both mathematical and practical real-world computations. Whether in engineering or pure mathematics, facilitating these simplifications is a foundational skill.