Find each product. Write answers in standard form. i(2+7i)(2-7i)

Verified step by step guidance

1. First, recognize that the expression involves the product of three factors: \(i\), \((2 + 7i)\), and \((2 - 7i)\).
2. Next, multiply the two complex conjugates \((2 + 7i)\) and \((2 - 7i)\) using the difference of squares formula: \((a + bi)(a - bi) = a^2 + b^2\). So, calculate \(2^2 + 7^2\).
3. Simplify the result from the previous step to get a real number.
4. Now multiply this real number by \(i\) to get the final product.
5. Express the final answer in standard form \(a + bi\), where \(a\) and \(b\) are real numbers.

Key Concepts

Complex Numbers and Imaginary Unit

Complex numbers have a real part and an imaginary part, expressed as a + bi, where i is the imaginary unit with the property i² = -1. Understanding how to manipulate i is essential for working with expressions involving complex numbers.

Multiplication of Complex Numbers

Multiplying complex numbers involves using the distributive property (FOIL method) and combining like terms, remembering that i² = -1. This process allows simplification of products into standard form a + bi.

Difference of Squares Formula

The expression (a + bi)(a - bi) uses the difference of squares formula: (x + y)(x - y) = x² - y². Applying this to complex conjugates simplifies the product to a real number, which is useful in simplifying complex expressions.