Problem 38 - Intermediate Algebra

Question

Multiply. Write the product in the form \(a+b i .\) See Example 4. $$ (\sqrt{5}-5 i)(\sqrt{5}+5 i) $$

Step-by-Step Solution

Answer

The product is \(30 + 0i\).

Step 1: Identify the Formula

Recognize the given expression \((\sqrt{5} - 5i)(\sqrt{5} + 5i)\) as a difference of squares pattern. The formula for the difference of squares is \((a-b)(a+b) = a^2 - b^2\). In this expression, \(a = \sqrt{5}\) and \(b = 5i\).

Step 2: Apply the Difference of Squares Formula

Apply the formula:\(a^2 - b^2 = (\sqrt{5})^2 - (5i)^2\). Calculate each square separately.

Step 3: Calculate the Squares

Calculate \((\sqrt{5})^2\). Since the square and square root cancel each other, \((\sqrt{5})^2 = 5\). Calculate \((5i)^2\). Recall that \((i)^2 = -1\), so \((5i)^2 = 25(i)^2 = 25(-1) = -25\).

Step 4: Combine the Results

Substitute these results back into the difference of squares equation: \(5 - (-25)\). Simplify to \(5 + 25 = 30\).

Step 5: Express the Answer in Standard Form

The result \(30\) can be written in the form \(a + bi\). Since there is no imaginary part, it is \(30 + 0i\).

Key Concepts

Difference of Squares

A difference of squares is a mathematical pattern that appears when you multiply two binomials of the form \(a-b\) and \(a+b\). You might wonder why it's called the "difference" of squares since you’re multiplying. Well, the trick is in observing how the squares involve subtraction once you apply the formula.

Recognize the pattern:

• \((a-b)(a+b) = a^2 - b^2\)

The \(a² - b²\) reflects a subtraction of perfect squares. Here, \(a\) and \(b\) are any numbers or expressions, including imaginary numbers like in our exercise.

This pattern is handy because instead of multiplying the whole expressions, you can square \(a\) and \(b\) separately, and then subtract the results. It's much faster and cuts down on potential errors.

Imaginary Numbers

Imaginary numbers are numbers that when squared, give a negative result. The core imaginary number is represented by \(i\), defined as the square root of \(-1\). This means that \(i^2 = -1\).

Why are they called 'imaginary'? At first, it seemed impossible to imagine a number resulting from a square root of a negative number. But now, they are quite real in mathematics and physics.

Key features:

• Used for solving equations that don’t have real solutions.
• Help in dealing with complex numbers (a combination of real and imaginary numbers).

For instance, in this exercise, \(5i\) represents 5 times the imaginary unit. So squaring it, as shown, gives a negative outcome. Understanding these numbers is crucial in unveiling more complex mathematical concepts.

Standard Form of Complex Numbers

Complex numbers combine both real numbers and imaginary numbers into a single expression. The standard form for representing complex numbers is \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part.

This form makes it easy to read and perform operations like addition or multiplication. In the solution to our problem, the complex number resolved to \(30 + 0i\), indicating that there was no imaginary component in the final result after all steps of simplification.

Benefits of using the standard form include:

• Clarity in distinguishing between real and imaginary components.
• Simplifying the computation of complex number operations.

Understanding the standard form helps in visualizing complex numbers on the complex plane—an essential aspect of mathematical analysis, engineering, and physics.