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

검증된 단계별 안내

First, recognize that the expression involves the product of three factors: \(i\), \((2 + 7i)\), and \((2 - 7i)\).

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 \(2^2 + 7^2\).

Simplify the result from the previous step to get a real number.

Now multiply this real number by \(i\) to get the final product.

Express the final answer in standard form \(a + bi\), where \(a\) and \(b\) are real numbers.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.