Problem 54 - Algebra 2

Question

Simplify. $$ \sqrt{-13} \cdot \sqrt{-26} $$

Step-by-Step Solution

Answer

\(-\sqrt{338}\)

Step 1: Understand the Expression

We are asked to simplify the expression \( \sqrt{-13} \cdot \sqrt{-26} \). Here, both numbers under the square root are negative, which means we will deal with imaginary numbers. The square root of a negative number involves the imaginary unit \( i \), where \( i = \sqrt{-1} \).

Step 2: Express the Square Roots Using the Imaginary Unit

Rewrite \( \sqrt{-13} \) as \( \sqrt{13} \cdot i \) and \( \sqrt{-26} \) as \( \sqrt{26} \cdot i \). So the expression becomes \( (\sqrt{13} \cdot i) \cdot (\sqrt{26} \cdot i) \).

Step 3: Simplify the Expression

Use the property \( i^2 = -1 \) to simplify. The expression \((\sqrt{13} \cdot i) \cdot (\sqrt{26} \cdot i) \) becomes \( \sqrt{13} \cdot \sqrt{26} \cdot i^2 \).

Step 4: Calculate the Square Roots and Multiply

Calculate the multiplication of the square roots: \( \sqrt{13} \cdot \sqrt{26} = \sqrt{13 \times 26} = \sqrt{338} \).

Step 5: Apply the Imaginary Unit Condition

Since \( i^2 = -1 \), multiply \( \sqrt{338} \) with \( i^2 \) to get \(-\sqrt{338} \). This is because multiplying by \( i^2 \) introduces the negative sign.

Key Concepts

Complex Numbers

Complex numbers are a fundamental concept in mathematics that expand our understanding beyond real numbers by introducing an imaginary component. A complex number is often expressed in the form \( a + bi \), where:

• \( a \) is the real part of the number
• \( b \) is the imaginary part, and
• \( i \) represents the imaginary unit, satisfying \( i^2 = -1 \).

This notation allows us to handle problems involving the square roots of negative numbers easily. Complex numbers are essential in many areas of mathematics, physics, and engineering because they enable calculations that would be impossible with just real numbers. Whenever you encounter a square root of a negative number, remember that you're diving into the exciting world of complex numbers.

Square Roots

The square root of a number \( x \) is a value that, when multiplied by itself, gives \( x \). For positive numbers, this is straightforward. However, the introduction of square roots of negative numbers calls for special treatment.
For instance, computing \( \sqrt{-13} \) involves using the imaginary unit \( i \) such that \( i = \sqrt{-1} \). The process follows these steps:

• Take the square root of the positive equivalent, \( \sqrt{13} \).
• Multiply it by \( i \) to express the negative number's square root as \( \sqrt{13} \cdot i \).

Using this method allows us to simplify expressions involving negative roots, such as \( \sqrt{-13} \cdot \sqrt{-26} \). We apply fundamental properties of square roots and the imaginary unit to arrive at a meaningful solution.

Properties of i

The imaginary unit \( i \) has unique properties that simplify working with square roots of negative numbers. Understanding these properties is essential:

• \( i = \sqrt{-1} \)
• \( i^2 = -1 \)
• \( i^3 = -i \)
• \( i^4 = 1 \), and the cycle repeats every four powers.

In the original problem, knowing that \( i^2 = -1 \) allowed us to simplify the expression further. When multiplying expressions that include \( i \), such as \( (\sqrt{13} \cdot i) \cdot (\sqrt{26} \cdot i) \), you can apply the property \( i^2 = -1 \) to introduce a negative sign. This step is crucial for finding the correct simplified form, resulting in \(-\sqrt{338}\) for the exercise provided.