In Exercises 1–10, perform the indicated operations and write the result in standard form. (7 + 8i)(7 − 8i)

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1. Recognize that the expression \((7 + 8i)(7 - 8i)\) is a product of two complex conjugates. The product of conjugates follows the formula \((a + bi)(a - bi) = a^2 + b^2\).
2. Identify the real part \(a = 7\) and the imaginary coefficient \(b = 8\) from the given expression.
3. Apply the formula by squaring the real part and the imaginary coefficient: calculate \(7^2\) and \(8^2\).
4. Add the squares together to get the result in the form \(a^2 + b^2\), which will be a real number since the imaginary parts cancel out.
5. Write the final answer in standard form \(x + yi\), where \(y = 0\) because the product of conjugates is always a real number.

Key Concepts

Complex Numbers and Standard Form

Complex numbers are expressed in the form a + bi, where a is the real part and b is the imaginary part. The standard form refers to writing the result explicitly as a sum of a real number and an imaginary number. Understanding this form is essential for interpreting and simplifying complex number operations.

Multiplication of Complex Conjugates

Multiplying a complex number by its conjugate (changing the sign of the imaginary part) results in a real number. This product equals the sum of the squares of the real and imaginary parts, i.e., (a + bi)(a - bi) = a² + b². This property simplifies calculations and eliminates imaginary components.

Use of the Imaginary Unit i

The imaginary unit i is defined by i² = -1. When multiplying complex numbers, powers of i must be simplified using this definition. Recognizing and applying i² = -1 allows conversion of imaginary terms into real numbers, facilitating the simplification of expressions.