Problem 51 - Algebra and Trigonometry

Question

Evaluate the radical expression, and express the result in the form \(a+b i\) $$ (3-\sqrt{-5})(1+\sqrt{-1}) $$

Step-by-Step Solution

Answer

The expression simplifies to \((3 + \sqrt{5}) + (3 - \sqrt{5})i\).

Step 1: Identify and Simplify Complex Numbers

First, identify the complex numbers in the expression \((3-\sqrt{-5})(1+\sqrt{-1})\). We know that \(\sqrt{-1} = i\), the imaginary unit, and \(\sqrt{-5} = \sqrt{5} \times i\). Thus, the expression becomes \((3 - \sqrt{5}i)(1 + i)\).

Step 2: Apply the Distributive Property

Use the distributive property (also known as the FOIL method for binomials) to expand the expression:\(3(1+i) - \sqrt{5}i(1+i)\).

Step 3: Distribute Each Term

Expand each part of the expression:- First, distribute \(3\): \(3 \times 1 = 3\) \(3 \times i = 3i\)- Next, distribute \(- \sqrt{5}i\): \(- \sqrt{5}i \times 1 = - \sqrt{5}i\) \(- \sqrt{5}i \times i = - \sqrt{5}i^2 = \sqrt{5}\) because \(i^2 = -1\).

Step 4: Combine Like Terms

Combine all the terms from the expansion:\[ 3 + 3i - \sqrt{5}i + \sqrt{5}. \]This simplifies to:\[ (3 + \sqrt{5}) + (3 - \sqrt{5})i. \]

Step 5: Express in Required Form

Express the result in the form \(a + bi\) where \(a = 3 + \sqrt{5}\) and \(b = 3 - \sqrt{5}\). The final expression is:\[ (3 + \sqrt{5}) + (3 - \sqrt{5})i. \]

Key Concepts

Imaginary Unit

In mathematics, the imaginary unit is denoted by the symbol \(i\). It is defined as \(i = \sqrt{-1}\). This is important because it allows us to work with numbers that are not real. When you encounter the square root of a negative number, like \(\sqrt{-5}\), it can be expressed in terms of \(i\) by recognizing that \(\sqrt{-1} = i\). Thus, \(\sqrt{-5}\) can be rewritten as \(\sqrt{5} \cdot i\), which is critical in simplifying expressions involving negative square roots. Recognizing the imaginary unit is key in working with complex numbers, which include both a real part and an imaginary part. When combined, they form expressions like \(a + bi\), where \(a\) and \(b\) are real numbers.

Distributive Property

The distributive property is a fundamental algebraic property used to multiply a single term by two or more terms inside a parenthesis. This property states that \(a(b + c) = ab + ac\). In the case of complex numbers, we often use this property to expand expressions. When applying the distributive property to binomials like \((3 - \sqrt{5}i)(1 + i)\), we expand each term by distributing one over the other. This process is also known as the FOIL method when specifically dealing with two binomials, where you multiply the First, Outer, Inner, and Last terms respectively. Breaking down expressions in this way helps to simplify and combine like terms easily.

Binomials

Binomials are algebraic expressions that contain exactly two terms, connected by either an addition or subtraction operator. For instance, expressions like \(3 - \sqrt{5}i\) and \(1 + i\) are binomials because they each have two distinct terms. When multiplying binomials, you typically apply the FOIL method which refers to multiplying the First terms together, the Outer terms, the Inner terms, and then the Last terms. This technique is quite helpful with complex numbers since it ensures each part of the expression is addressed and calculated, resulting in a more manageable form after combining and simplifying the terms. Understanding how to manipulate binomials is crucial for tackling more complicated algebraic tasks.

Simplification

Simplification involves reducing an expression to its most basic form. In the context of complex numbers, this often means combining like terms and acknowledging properties like \(i^2 = -1\). For example, when you expand the expression \((3 - \sqrt{5}i)(1 + i)\), you break it into components like \(3 + 3i - \sqrt{5}i + \sqrt{5}\). You then simplify this by combining the imaginary parts and the real parts: \((3 + \sqrt{5}) + (3 - \sqrt{5})i\). Noticing that \(-\sqrt{5}i \cdot i\) equals \(\sqrt{5}\) is a crucial part of simplification, thanks to the property \(i^2 = -1\). Simplification helps make expressions easier to interpret and use in further calculations.