Problem 81 - Intermediate Algebra

Question

Find each of the products and express the answers in the standard form of a complex number. $$ (6+7 i)(6-7 i) $$

Step-by-Step Solution

Answer

The product is 85 + 0i.

Step 1: Identify the Expression

The expression we need to find the product for is \((6 + 7i)(6 - 7i)\). This is a product of two complex conjugates.

Step 2: Recognize the Pattern

Use the pattern for the product of two complex conjugates: \((a + bi)(a - bi) = a^2 + b^2\). Here, \(a = 6\) and \(b = 7\).

Step 3: Apply the Pattern

Apply the formula to find the product: \(a^2 + b^2 = 6^2 + 7^2\).

Step 4: Calculate the Squares

Calculate each component: \(6^2 = 36\) and \(7^2 = 49\).

Step 5: Add the Results

Add the squared results: \(36 + 49 = 85\).

Step 6: Express in Standard Form

The standard form of a complex number is \(a + bi\), where \(a\) and \(b\) are real numbers. Since there is no imaginary component in the final result, the standard form is simply \(85 + 0i\).

Key Concepts

Complex Conjugates

When dealing with complex numbers, the concept of complex conjugates is very important. A complex conjugate is formed by changing the sign of the imaginary part of a complex number. For example, if you have the complex number \(a + bi\), its conjugate would be \(a - bi\). This operation is useful because when you multiply a complex number by its conjugate, the imaginary parts cancel out, leaving a real number.

• For \((6 + 7i)\), the conjugate is \((6 - 7i)\).
• The multiplication of these conjugates results in a product where the imaginary parts are neutralized.

Recognizing complex conjugates helps simplify expressions and solve equations involving complex numbers, as seen in the original exercise.

Standard Form

The standard form of a complex number is expressed as \(a + bi\). In this format, \(a\) is the real part and \(b\) is the imaginary part. It’s important because it gives a clear structure to complex numbers, making it easier to perform arithmetic operations and understand their properties.

• In the context of our problem, after applying the formula for complex conjugates, we arrive at \(85 + 0i\).
• This shows that the result of our multiplication is purely real, as there is no imaginary component left.

Understanding the standard form is crucial for working with complex numbers and effectively communicating your results.

Algebraic Expressions

Algebraic expressions in complex numbers follow the same basic principles as those in real numbers, but you must handle the imaginary unit \(i\), which satisfies \(i^2 = -1\). This adds an extra layer when performing arithmetic operations like addition, subtraction, multiplication, and division.

• Multiplying \((6 + 7i)\) and \((6 - 7i)\) involves distributing each component, similar to real numbers, but requires special attention to \(i^2\).
• In our example, using the pattern \((a + bi)(a - bi) = a^2 + b^2\), simplifies computation by avoiding direct use of \(i^2 = -1\).

Mastering the manipulation of algebraic expressions with complex numbers allows for a deeper understanding of mathematical structures and the ability to solve more complex problems.