Perform each operation. Write answers in standard form. -5i(3-i)²

검증된 단계별 안내

First, recognize that the problem requires you to simplify the expression \(-5i(3 - i)^2\) and write the answer in standard form, which is \(a + bi\) where \(a\) and \(b\) are real numbers.

Start by expanding the squared term \((3 - i)^2\). Use the formula for squaring a binomial: \((a - b)^2 = a^2 - 2ab + b^2\). Here, \(a = 3\) and \(b = i\), so write out the expansion as \(3^2 - 2 \times 3 \times i + i^2\).

Calculate each part of the expansion: \(3^2 = 9\), \(-2 \times 3 \times i = -6i\), and recall that \(i^2 = -1\). Substitute these values back into the expression to get \(9 - 6i + (-1)\).

Combine the real terms \(9\) and \(-1\) to simplify the expression inside the parentheses to \(8 - 6i\). Now, multiply this result by \(-5i\), so you have \(-5i(8 - 6i)\).

Distribute \(-5i\) across both terms inside the parentheses: multiply \(-5i \times 8\) and \(-5i \times (-6i)\). Remember to use the property \(i^2 = -1\) when simplifying the product involving \(i^2\). After this, combine like terms to write the expression in the form \(a + bi\).

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주요 개념

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Complex Numbers and Imaginary Unit

Complex numbers consist of 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 simplifying expressions involving imaginary numbers.

Exponentiation of Binomials

Raising a binomial to a power, such as (3 - i)², involves applying the distributive property or the formula (a - b)² = a² - 2ab + b². This step is crucial for expanding and simplifying the expression before further operations.

Multiplication of Complex Numbers

Multiplying complex numbers requires distributing each term and combining like terms, remembering to replace i² with -1. This process helps in simplifying the product into standard form a + bi.