In Exercises 9–20, find each product and write the result in standard form. −3i(7i − 5)

검증된 단계별 안내

Recall that the imaginary unit \(i\) has the property \(i^2 = -1\).

Distribute \(-3i\) across the terms inside the parentheses: \(-3i \times 7i\) and \(-3i \times (-5)\).

Calculate each product separately: \(-3i \times 7i = -21i^2\) and \(-3i \times (-5) = 15i\).

Substitute \(i^2\) with \(-1\) in the expression \(-21i^2\) to get \(-21 \times (-1)\).

Simplify the expression to combine the real and imaginary parts, resulting in a number in the form \(a + bi\).

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주요 개념

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

Imaginary Unit (i)

The imaginary unit 'i' is defined as the square root of -1, with the property that i² = -1. It is fundamental in complex numbers and allows for the extension of the real number system to include solutions to equations like x² + 1 = 0.

Multiplication of Complex Numbers

Multiplying complex numbers involves using the distributive property (FOIL method) and applying the rule i² = -1. Each term is multiplied carefully, combining like terms and simplifying to express the result in standard form a + bi.

Standard Form of a Complex Number

The standard form of a complex number is written as a + bi, where 'a' is the real part and 'b' is the coefficient of the imaginary part. Expressing results in this form clearly separates real and imaginary components for easier interpretation.