Problem 47 - Algebra and Trigonometry

Question

Evaluate the radical expression, and express the result in the form \(a+b i\) $$ \sqrt{-25} $$

Step-by-Step Solution

Answer

\(\sqrt{-25} = 0 + 5i\)

Step 1: Understand the Problem

The problem asks us to evaluate the radical expression \(\sqrt{-25}\) and express it in the form \(a + bi\), where \(a\) and \(b\) are real numbers and \(i\) is the imaginary unit.

Step 2: Simplify the Radical

Since \(\sqrt{-25}\) involves a negative number under the square root, it means we can express this in terms of the imaginary unit. Recall that \(i = \sqrt{-1}\). Therefore, \(\sqrt{-25} = \sqrt{25} \times \sqrt{-1} = 5i\).

Step 3: Write the Expression in the Form \(a+bi\)

The expression \(5i\) can be rewritten in the form \(a+bi\) as \(0 + 5i\). Here, \(a = 0\) and \(b = 5\).

Key Concepts

Imaginary Unit

The imaginary unit, denoted as \( i \), is a key concept in complex numbers. It is defined by the property \( i^2 = -1 \). This unique idea allows us to work with the square roots of negative numbers, which is not possible within the real number system.

When you come across a square root that involves a negative sign, like \( \sqrt{-1} \), the imaginary unit provides a solution: \( i \). This enables us to tackle mathematical problems involving radicals of negative numbers without confusion. As illustrated in the given exercise, to evaluate \( \sqrt{-25} \), we used \( i \) to express it as \( 5i \). In this result, 5 is the real part, scaled by \( i \).

Remembering the properties of \( i \) can help simplify complex expressions and identify the imaginary component of a radical expression.

Radical Expressions

Radical expressions involve roots, especially square roots. When working with these expressions, encountering a negative number underneath the radical sign might seem perplexing at first. However, complex numbers allow us to deal with such situations gracefully.

In real-world terms, dealing with the square root of a negative number would be a dead end because no real number multiplied by itself gives a negative product. But using the imaginary unit facilitates a new form of expression. The step-by-step solution you see for \( \sqrt{-25} \) showcases exactly how this works: treat \(-25\) as \( 25 \times -1 \), separate them into \( \sqrt{25} \) and \( \sqrt{-1} \), and you're left with \( 5i \).

Breaking down radical expressions into real and imaginary parts, through use of the imaginary unit, can give clarity and reveal the full value of expressions that might otherwise seem unsolvable.

Complex Form

Complex form in mathematics is represented as \(a + bi\), where \(a\) and \(b\) are real numbers and \(i\) is the imaginary unit. This form is essential to representing numbers that contain both real and imaginary components.

The real part, \(a\), signifies the horizontal distance on a Cartesian plane, while the imaginary part, \(b\), represents the vertical distance. For the expression \(5i\), seen in the original problem, it is written in complex form as \(0 + 5i\). Here:\

• \(a = 0\), implying there is no horizontal component; we are solely on the imaginary axis.
• \(b = 5\), indicating the magnitude along the imaginary axis.

Understanding complex form aids in visualizing and computing calculations in both algebra and applicable real-life scenarios. It allows an integrated approach to expressions that encompass both real and non-real components.