Problem 27 - Intermediate Algebra

Question

Express each number in terms of \(i\). $$ \sqrt{-\frac{25}{9}} $$

Step-by-Step Solution

Answer

\(\sqrt{-\frac{25}{9}} = \frac{5i}{3}\)

Step 1: Identify the imaginary unit

The imaginary unit is represented by \(i\), which is defined as \(i = \sqrt{-1}\). This is crucial for converting square roots of negative numbers into terms of \(i\).

Step 2: Simplify the expression under the square root

The expression given is \(\sqrt{-\frac{25}{9}}\). Start by recognizing that \(-\frac{25}{9} = \frac{-25}{9}\). This helps in simplifying expressions individually.

Step 3: Separate the fraction into its components

Take the square root of each part of the fraction separately: \(\sqrt{-\frac{25}{9}} = \sqrt{\frac{-25}{9}} = \sqrt{-25} \cdot \sqrt{\frac{1}{9}}\).

Step 4: Simplify each part separately

For \(\sqrt{-25}\), rewrite this as \(\sqrt{-1} \cdot \sqrt{25}\). Using the definition of \(i\), this becomes \(i \cdot 5 = 5i\). For \(\sqrt{\frac{1}{9}}\), note that this is \(\frac{1}{3}\) since \(9 = 3^2\).

Step 5: Combine the results

Now, combine the simplified results: \(\sqrt{-\frac{25}{9}} = \frac{5i}{3}\). This expresses the original expression in terms of \(i\).

Key Concepts

Imaginary Unit

The imaginary unit, denoted by \(i\), is a fundamental concept in complex numbers. It's defined as \(i = \sqrt{-1}\). This means \(i^2 = -1\). The imaginary unit allows us to evaluate square roots of negative numbers, which is not possible within the realm of real numbers.

In real numbers, the square root function can only take non-negative numbers. However, \(i\) opens the door to a broader number system known as complex numbers. In this system, expressions like \(\sqrt{-25}\) can be calculated and expressed as \(5i\). Usages of \(i\) extend beyond mathematics, influencing engineering, physics, and computer science, where systems require calculations involving square roots of negative values.

Square Root of Negative Numbers

The ability to find a square root for negative numbers relies on the use of the imaginary unit \(i\). When you encounter a negative number under a square root, you can separate it into its positive component and the imaginary unit.

For example, consider \(\sqrt{-25}\). This can be broken down by considering it as \(\sqrt{-1 \times 25} = \sqrt{-1} \times \sqrt{25}\).

• The \(\sqrt{-1}\) is simply \(i\), the imaginary unit.
• The square root of 25 is 5.

Thus, \(\sqrt{-25} = 5i\), a representation that maintains the integrity of mathematical operations while extending it into the complex number system.

Whenever you encounter this scenario, remember it's about reorganizing the expression to separate the negative from its square root and leverage \(i\) as the bridge.

Simplifying Fractions with Square Roots

Simplifying fractions that have square roots can seem tricky at first, especially when negatives are involved. But by breaking it down into manageable steps, it becomes clearer.

Let's simplify \(\sqrt{-\frac{25}{9}}\). We start by separating the negative and the fractions:

• Write it as \(\sqrt{-25} \times \sqrt{\frac{1}{9}}\).
• Solve each square root separately. \(\sqrt{-25} = 5i\), because \(-25\) becomes \(-1 \times 25\) and \(\sqrt{-1} = i\).
• For \(\sqrt{\frac{1}{9}}\), recognize it as \(\frac{1}{3}\), since 9 is a perfect square \((3^2)\).

Now, combining the separate results gives you \(\frac{5i}{3}\). This approach of tackling each part individually simplifies the complexity, making these expressions easier to handle in terms of the imaginary unit.