Problem 42 - College Algebra and Calculus: An Applied Approach

Question

Perform the indicated operation and write the result in standard form. $$(\sqrt{14}+\sqrt{10} i)(\sqrt{14}-\sqrt{10} i)$$

Step-by-Step Solution

Answer

The result of the operation in standard form is 24.

Step 1 - Identify Elements

Identify the two numbers to multiply. They are \((\sqrt{14}+\sqrt{10} i)\) and \((\sqrt{14}-\sqrt{10} i)\). Both are complex numbers, a + bi form, where a and b are real numbers and i is the imaginary unit.

Step 2 - Apply Difference of Squares

We have two complex numbers, where the second one is the conjugate of the first one. We can apply the difference of squares formula here, that is (a+b)(a-b) becomes \(a^2 - b^2\). Therefore, our result is \((\sqrt{14})^2 - (\sqrt{10} i)^2\).

Step 3 - Simplify the Expression

Simplify the expression obtained from Step 2. The given expression simplifies to \(14 - 10i^2\). Remember that \(i^2\) equals to -1.

Step 4 - Final Step

- Substitute -1 for \(i^2\). This results in \(14 - 10(-1)\), or \(14 + 10\), which equals to 24.

Key Concepts

Imaginary Unit

In the realm of complex numbers, understanding the imaginary unit, denoted as \(i\), is essential. The imaginary unit is defined as the square root of -1. It is a fundamental concept used to extend real numbers into complex numbers, allowing mathematicians and engineers to solve equations that do not have real solutions.
For example, the equation \(x^2 + 1 = 0\) has no real solution since no real number squared results in -1. Introducing \(i\) as \(i^2 = -1\), allows us to express solutions using complex numbers, revealing new results and insights into mathematical problems.
In the context of our exercise, the term \(\sqrt{10}i\) utilizes the imaginary unit. When we multiply or perform operations on complex numbers, it's crucial to remember that \(i^2\) simplifies to -1. This simplification is applied in step 3 of our solution to transition our complex number multiplication into the standard form of a real number.

Difference of Squares

The difference of squares is a useful algebraic identity that appears frequently in mathematics. It states that \((a+b)(a-b) = a^2 - b^2\). This identity simplifies the product of two conjugates into the difference of their individual squares.
Conjugates are pairs of expressions like \( (a+bi) \) and \( (a-bi) \) in complex numbers. These conjugates, when multiplied, utilize the difference of squares identity by removing the imaginary parts, simplifying the expression directly to a real number.
In our specific exercise, we have two numbers \((\sqrt{14} + \sqrt{10}i)\) and its conjugate \((\sqrt{14} - \sqrt{10}i)\). By applying the difference of squares, we simplify this to \((\sqrt{14})^2 - (\sqrt{10}i)^2\), significantly reducing the complexity of our calculation and quickly converging towards a result in its standard form.

Standard Form

Complex numbers are typically expressed in the standard form \(a + bi\), where \(a\) and \(b\) are real numbers and \(i\) is the imaginary unit. The standard form provides a clear way to identify the real and imaginary components of a complex number.
During the solution of our exercise, the process is aimed at transforming the multiplication of two complex numbers, a non-standard representation, into a real number in standard form where \(b = 0\). By following the steps of simplification, utilizing both the imaginary unit properties and difference of squares, we arrive at a result where our expression has been simplified to \(24 + 0i\), further reducing to \(24\).
This notation isn't merely a simplification—it provides a more succinct understanding of complex numbers and is widely used in mathematical, engineering, and scientific contexts to communicate solutions efficiently.