Problem 50 - Algebra and Trigonometry

Question

Evaluate the expression and write the result in the form \(a+b i .\) $$ (\sqrt{3}-\sqrt{-4})(\sqrt{6}-\sqrt{-8}) $$

Step-by-Step Solution

Answer

The expression evaluates to \(-\sqrt{2} - 4\sqrt{6}i\).

Step 1: Simplify Complex Numbers

The given expression is \((\sqrt{3} - \sqrt{-4})(\sqrt{6} - \sqrt{-8})\). First, simplify the complex numbers: \(\sqrt{-4} = 2i\) and \(\sqrt{-8} = \sqrt{4 \times -2} = 2\sqrt{2}i\). Thus, the expression becomes \((\sqrt{3} - 2i)(\sqrt{6} - 2\sqrt{2}i)\).

Step 2: Expand the Product

Use the distributive property to expand: \((\sqrt{3} - 2i)(\sqrt{6} - 2\sqrt{2}i) = \sqrt{3}\sqrt{6} - \sqrt{3}\cdot 2\sqrt{2}i - 2i\sqrt{6} + 2i\cdot 2\sqrt{2}i\).

Step 3: Calculate Each Term

Calculate each term individually:- \(\sqrt{3}\sqrt{6} = \sqrt{18} = 3\sqrt{2}\).- \(\sqrt{3} \cdot 2\sqrt{2}i = 2\sqrt{6}i\).- \(2i\sqrt{6} = 2\sqrt{6}i\).- \(2i \cdot 2\sqrt{2}i = 4\sqrt{2}i^2 = -4\sqrt{2}\) because \(i^2 = -1\).

Step 4: Combine Terms

Combine the terms:- The real parts: \(3\sqrt{2} - 4\sqrt{2} = -\sqrt{2}\).- The imaginary parts: \(-2\sqrt{6}i - 2\sqrt{6}i = -4\sqrt{6}i\).Thus, the expression in standard form is \(-\sqrt{2} - 4\sqrt{6}i\).

Key Concepts

Imaginary Unit

The imaginary unit, often denoted as "i," is a mathematical concept that extends the real number system. It is defined by the property that its square is -1. In mathematical terms, this is expressed as:

• \( i^2 = -1 \)

When complex numbers involve the square root of a negative number, the imaginary unit allows us to express this operation. For example, \( \sqrt{-4} \) simplifies to \( 2i \) because \( i \) serves as a factor to convert the negative under the square root into a product with the imaginary unit.

The imaginary unit "i" is the foundation upon which all complex numbers are built, and it facilitates the inclusion of solutions that are otherwise not possible in the real number system, such as solving polynomial equations that have no real roots.

Distributive Property

The distributive property in mathematics is a powerful tool used to simplify expressions and perform calculations efficiently. This property allows us to multiply a single term by each term in a bracketed expression. The formula for the distributive property is:

• \( a(b+c) = ab + ac \)

In the context of multiplying complex numbers, the distributive property is used to expand expressions such as

• \((\sqrt{3} - 2i)(\sqrt{6} - 2\sqrt{2}i) \)

in the original exercise:

• \( \sqrt{3}\sqrt{6} - \sqrt{3}\cdot 2\sqrt{2}i - 2i\sqrt{6} + 2i\cdot 2\sqrt{2}i \)

By distributing each term across the parentheses, complex multiplication becomes simpler and each constituent part of the expression is calculated individually, before being summed to form the final result.

Complex Conjugate

The complex conjugate involves flipping the sign of the imaginary component of a complex number. For a complex number \(a + bi\), its complex conjugate is \(a - bi\). The concept of a complex conjugate is important because:

• It is used to simplify division in complex numbers.
• Complex conjugates help in computing the modulus (or absolute value) of a complex number, as multiplying a complex number by its conjugate results in a real number.

While the exercise provided doesn't explicitly use complex conjugates, understanding them is crucial in operations involving complex numbers, particularly in simplifying results and ensuring expressions remain in standard form.

Imaginary Numbers

Imaginary numbers are an extension of the real number system, essential for solving equations that involve the square roots of negative numbers. An imaginary number can be expressed as a real number multiplied by the imaginary unit \(i\). For instance:

• \( \sqrt{-4} \) becomes \( 2i\).

Imaginary numbers, when combined with real numbers, form complex numbers (\(a + bi\)) which are widely used in engineering and physics. They allow for the full spectrum of solutions in quadratic equations and other polynomials to be explored, beyond what is possible with only real numbers.

• Imaginary numbers are crucial for expressing the roots of negative numbers.
• They are used extensively in computational fields and quantum physics.

In summary, imaginary numbers bring depth to our understanding of mathematical concepts, extending our ability to model and compute within diverse scientific contexts.