Problem 31 - Algebra and Trigonometry Real Mathematics, Real People

Question

Perform the operation and write the result in standard form. $$\sqrt{-6} \cdot \sqrt{-2}$$

Step-by-Step Solution

Answer

\(-2\sqrt{3}\)

Step 1: Express the Given Numbers as Complex Numbers

Express \( \sqrt{-6} \) and \( \sqrt{-2} \) as \( i\sqrt{6} \) and \( i\sqrt{2} \) respectively.

Step 2: Perform the Multiplication

Multiply the two numbers: \( i\sqrt{6} \) and \( i\sqrt{2} \) to get \( -\sqrt{12} \) or \( -2\sqrt{3} \). This is because \( i \cdot i = -1 \) and \( \sqrt{6} \cdot \sqrt{2} = \sqrt{12} \).

Step 3: Convert to Standard Form

Express the solution \( -2\sqrt{3} \) in the standard form, which is the simplest form to present this number. So it remains the same.

Key Concepts

Imaginary Numbers

Sometimes, when you try to take the square root of a negative number, you get what we call an imaginary number. Imaginary numbers are built on the idea that no real number squared will give you a negative result. So, mathematicians introduced the imaginary unit, denoted as \( i \), where \( i^2 = -1 \). Anything that involves \( \sqrt{-1} \) can be written as \( i \).
For example:

• \( \sqrt{-6} = i\sqrt{6} \)
• \( \sqrt{-2} = i\sqrt{2} \)

This transformation allows us to work with what were previously unsolvable problems using real numbers. Essentially, it lets us "move into" the imaginary world where these calculations make sense.

Square Roots

The square root is usually about finding a number, which when multiplied by itself, gives the original number. But what happens when the number is negative? That's where the imaginary unit \( i \) comes into play.

Notice how \( \sqrt{-6} \) and \( \sqrt{-2} \) can become \( i\sqrt{6} \) and \( i\sqrt{2} \). This trick allows us to convert a negative inside a square root into an imaginary number, with \( i \) representing \( \sqrt{-1} \).

Working with imaginary numbers involves recognizing these transformations ensure calculations remain valid:

• Avoid leaving a negative under a square root.
• Don't forget, \( i \cdot i = -1 \), which will be necessary for operations like multiplying imaginary numbers.

This converts a complicated problem into one with a straightforward approach where you can multiply real parts and imaginary parts separately.

Multiplication of Complex Numbers

When you multiply complex numbers, the process is similar to multiplying binomials. Consider numbers in the form \( a + bi \), where \( i \) is the imaginary unit. The key step is following the distributive property, often remembered by the acronym FOIL (First, Outer, Inner, Last).

In this problem, we multiply \( i\sqrt{6} \) and \( i\sqrt{2} \):

• First, multiply the real components: \( \sqrt{6} \times \sqrt{2} = \sqrt{12} \).
• Then, multiply the imaginary components: \( i \times i = -1 \).

Combining these results, we have \( -\sqrt{12} \), simplifying to \( -2\sqrt{3} \). This example demonstrates how multiplying imaginary parts can introduce negative numbers, transforming complex expressions into simpler expressions, or even real numbers. Always break down each step and clearly address both the real and imaginary components separately for accurate multiplication.