Problem 52 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The result is \(3\sqrt{2} - 8 - 4\sqrt{6}i\).

Step 1: Write the expression with imaginary unit

Rewrite the given expression using the imaginary unit \(i\), where \(i = \sqrt{-1}\). The expression \(\sqrt{-4}\) becomes \(\sqrt{4} \cdot i = 2i\) and \(\sqrt{-8}\) becomes \(\sqrt{8} \cdot i = 2\sqrt{2}i\). Then the expression is \((\sqrt{3} - 2i)(\sqrt{6} - 2\sqrt{2}i)\).

Step 2: Expand the expression using the distributive property

Apply the distributive property: \((a - b)(c - d) = ac - ad - bc + bd\). For our expression, it becomes:\[(\sqrt{3} \cdot \sqrt{6}) - (\sqrt{3} \cdot 2\sqrt{2}i) - (2i \cdot \sqrt{6}) + (2i \cdot 2\sqrt{2}i)\].

Step 3: Simplify real and imaginary terms

Calculate each term separately:- \(\sqrt{3} \cdot \sqrt{6} = \sqrt{18} = 3\sqrt{2}\)- \(\sqrt{3} \cdot 2\sqrt{2}i = 2\sqrt{6}i\)- \(2i \cdot \sqrt{6} = 2\sqrt{6}i\)- \(2i \cdot 2\sqrt{2}i = -8\) (because \(i^2 = -1\))Combine like terms: \(3\sqrt{2} - 4\sqrt{6}i - 8\).

Step 4: Rearrange into the form \(a + bi\)

The expression is simplified to \(3\sqrt{2} - 8 - 4\sqrt{6}i\). Combine the real terms to get it in the form \(a + bi\), which is:\[(3\sqrt{2} - 8) + (-4\sqrt{6})i\].

Key Concepts

Radical Expressions

Radical expressions involve roots such as square roots or cube roots. In this exercise, we encounter expressions like \(\sqrt{3}\) and \(\sqrt{-4}\). These symbols tell us we are looking for a number, which when multiplied by itself gives us the number inside the radical sign. For instance, \(\sqrt{3}\) specifies a number that, when multiplied by itself, equals 3.

• Radicals can be converted to exponential form, such as \(\sqrt{a} = a^{1/2}\), which can assist in simplifying expressions.
• The root of a negative number, like \(\sqrt{-4}\), requires using the imaginary unit \(i\) because square roots of negatives are not real.

Understanding how to manipulate these radicals is essential for working through complex expressions, especially as we look to simplify them with imaginary numbers.

Imaginary Unit

The imaginary unit, usually represented by the symbol \(i\), is a mathematical concept that allows us to work with the square roots of negative numbers. By definition, \(i\) is equal to \(\sqrt{-1}\). So when we encounter \(\sqrt{-4}\), we can express it as \(\sqrt{4} \cdot i\), which simplifies to \(2i\).

• This notion transforms negative radicals into manageable terms by isolating the negative aspect as \(i\).
• Using \(i\), mathematicians can explore complex numbers, which are written as \(a + bi\), where \(a\) and \(b\) are real numbers.

Applying \(i\) helps keep track of these components, especially during multiplication and simplification.

Distributive Property

The distributive property is a basic yet powerful concept in algebra used to break down expressions for easier calculation. It describes how to multiply a single term across a sum or difference within parentheses. In its simplest form, it can be written as \(a(b + c) = ab + ac\).
For example, in the original expression \((\sqrt{3} - 2i)(\sqrt{6} - 2\sqrt{2}i)\), the distributive property guides us to multiply everything inside the parentheses, resulting in several terms:

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

This approach is essential for working with polynomials and complex number expressions, ensuring all parts of the expression are correctly dealt with.

Simplification of Expressions

Simplifying expressions means reducing them to their most basic form. In our context, this involves combining like terms and reducing complex number expressions. After using the distributive property, we obtain several terms that can often be simplified and combined.

• Identify and calculate each term, starting with simplifying square roots and handling the imaginary unit \(i\).
• Combine like terms, such as adding or subtracting real components separately from imaginary components.
• Always aim to reduce expressions to the form \(a + bi\), where \(a\) is the real part and \(b\) is the coefficient of the imaginary part.

In this exercise, simplification first involves calculating terms like \(\sqrt{3} \cdot \sqrt{6} = 3\sqrt{2}\) and then ensuring the result is neatly expressed as a complex number.