Problem 51 - Precalculus Mathematics for Calculus

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 result is \((3 + \sqrt{5}) + (3 - \sqrt{5})i\).

Step 1: Identify the imaginary components

Rewrite the radicals with imaginary units where applicable. We have \( \sqrt{-5} = \sqrt{5} \cdot i \) and \( \sqrt{-1} = i \). So the expression becomes: \((3 - \sqrt{5}i)(1 + i)\)

Step 2: Distribute the terms

Use the distributive property (FOIL – First, Outer, Inner, Last) to expand the expression. Calculate:\[3(1) + 3(i) - (\sqrt{5}i)(1) - (\sqrt{5}i)(i) \]This simplifies to:\[3 + 3i - \sqrt{5}i - \sqrt{5}i^2\]

Step 3: Simplify using \(i^2 = -1\)

Since \(i^2 = -1\), substitute this in:\[3 + 3i - \sqrt{5}i - \sqrt{5}(-1)\]This simplifies to:\[3 + 3i - \sqrt{5}i + \sqrt{5}\]

Step 4: Combine like terms

Group and combine the real and imaginary parts of the expression:\[(3 + \sqrt{5}) + (3i - \sqrt{5}i)\]Combine the imaginary parts:\[3i - \sqrt{5}i = (3 - \sqrt{5})i\]Thus, the expression simplifies to:\[(3 + \sqrt{5}) + (3 - \sqrt{5})i\]

Key Concepts

Imaginary Unit

Complex numbers often involve the imaginary unit, denoted as \(i\). In mathematics, \(i\) is defined by the property \(i^2 = -1\). This is a crucial concept because there is no real number that satisfies this condition, making \(i\) an essential part of complex numbers.

• When you encounter a negative number under a square root, like \(\sqrt{-1}\), it transforms into \(i\).
• Similarly, \(\sqrt{-5}\) can be written as \(\sqrt{5} \cdot i\).

Thus, using the imaginary unit allows us to handle such negative square roots, expanding the possibilities of solving complex equations. Complex numbers are then expressed in the form \(a + bi\), where \(a\) and \(b\) are real numbers. Understanding \(i\) is crucial when working with complex expressions and equations.

Distributive Property

The distributive property is an essential mathematical principle that comes into play when you need to multiply two expressions. In the context of complex numbers, this is often referred to when expanding expressions like \((3 - \sqrt{5}i)(1 + i)\).

• The distributive property, in simple terms, states \(a(b+c) = ab + ac\).
• In more complex forms, it extends to using the FOIL method (First, Outside, Inside, Last) to ensure each part of one binomial multiplies by each part of the second.

For our specific example:

• First: Multiply the first terms of each binomial, \(3 \times 1 = 3\).
• Outside: Multiply the outer terms, \(3 \times i = 3i\).
• Inside: Multiply the inside terms, \(-\sqrt{5}i \times 1 = -\sqrt{5}i\).
• Last: Multiply the last terms, \(-\sqrt{5}i \times i = -\sqrt{5}i^2\).

This thorough process ensures all components are accounted for, aiding significantly in solving expressions accurately.

Simplifying Radicals

Simplifying radicals is about making expressions involving roots easier to manage and understand. It especially aids in streamlining solutions when working with complex numbers in expressions like \(\sqrt{-5}\).

• When simplifying \(\sqrt{-5}\), recognize it as \(\sqrt{5} \cdot i\) because of the imaginary unit \(i\).
• Similarly, when you have \(\sqrt{5}(-1) = -\sqrt{5}\), you might need to apply the property \(i^2 = -1\).

In problem-solving, simplifying radicals helps make expressions less cluttered and reduces complexity. After distributing and simplifying these radicals, the next step is to combine like terms, isolating real parts from imaginary parts for a clean, easy-to-understand result. Following these steps, efforts in reducing radicals are not just about simplification but also about increasing clarity.