Problem 10 - Algebra A Combined Function

Question

Multiply or divide as indicated. $$ \sqrt{-11} \cdot \sqrt{-3} $$

Step-by-Step Solution

Answer

The result is \(-\sqrt{33}\).

Step 1: Identify the Components

The expression given is \( \sqrt{-11} \cdot \sqrt{-3} \). Notice that both components under the square roots are negative numbers, \(-11\) and \(-3\).

Step 2: Use the Property of Imaginary Numbers

The square root of a negative number can be expressed using the imaginary unit \(i\), where \(i = \sqrt{-1}\). Thus, \(\sqrt{-11} = \sqrt{11} \cdot i\) and \(\sqrt{-3} = \sqrt{3} \cdot i\).

Step 3: Rewrite the Expression with Imaginary Units

Substitute the expressions from Step 2 into the original problem: \( (\sqrt{11} \cdot i) \cdot (\sqrt{3} \cdot i) \).

Step 4: Multiply the Radicals and Imaginary Units Separately

Combine the radicals and the imaginary units: \((\sqrt{11} \cdot \sqrt{3}) \cdot (i \cdot i) = \sqrt{33} \cdot i^2\).

Step 5: Simplify Using the Property of \(i^2\)

Since \(i^2 = -1\), replace \(i^2\) with \(-1\): \(\sqrt{33} \cdot (-1) = -\sqrt{33}\).

Step 6: Conclusion: Final Expression

The result of multiplying \( \sqrt{-11} \cdot \sqrt{-3} \) is \(-\sqrt{33}\).

Key Concepts

Complex Numbers

Complex numbers are fascinating in the world of mathematics because they extend what we know about real numbers. A complex number is expressed in the form of \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) represents the imaginary unit. This combination of real and imaginary parts enables us to solve equations that have no solutions in the real number system alone. For example, the equation \(x^2 + 1 = 0\) has no real solution because there is no real number whose square is negative. However, using complex numbers, the solution is possible.

• Imaginary Part: The "\(bi\)" in a complex number represents the imaginary component, where \(i = \sqrt{-1}\).
• Real Part: The "\(a\)" is just a regular real number.

The introduction of complex numbers has expanded the realm of mathematical solutions, offering a more comprehensive understanding of certain patterns in numbers and functions.

Square Roots

Square roots are numbers that produce a specified value when multiplied by themselves. In the context of the problem \(\sqrt{-11} \cdot \sqrt{-3}\), square roots of negative numbers are involved, which cannot be solved within the real number framework.
To tackle square roots of negative numbers, we turn to imaginary numbers. The key is the imaginary unit \(i\), defined as \(\sqrt{-1}\). With this unit in hand, we can express any negative number's square root. For example:

• \(\sqrt{-11} = \sqrt{11} \cdot i\)
• \(\sqrt{-3} = \sqrt{3} \cdot i\)

This method allows us to manage square roots of negative numbers and perform operations like multiplication, as in the exercise, and simplifies expressions to be solved.

Properties of i

The imaginary unit \(i\) has unique properties that make it indispensable in mathematics, particularly in dealing with complex numbers. The core property to remember is that \(i\) squared equals -1:

• \(i = \sqrt{-1}\)
• \(i^2 = -1\)

These properties allow for simplified calculations involving complex numbers. For example, when multiplying you often deal with \(i^2\), which should directly be replaced with \(-1\) in calculations. In solving \(\sqrt{-11} \cdot \sqrt{-3}\), after expressing it in terms of \(i\) and multiplying, we used \(i^2 = -1\) to reach the solution \(-\sqrt{33}\).
This process underscores the elegance of imaginary numbers as tools for solving problems that real numbers alone cannot address. Understanding these properties equips students to navigate further challenges involving complex expressions.