Find each product and write the result in standard form. (2 + 3i)²

검증된 단계별 안내

Recall that to find the product of a complex number squared, such as $(2 + 3i)^2$, you can use the formula for the square of a binomial: $(a + b)^2 = a^2 + 2ab + b^2$.

Identify $a = 2$ and $b = 3i$ in the expression $(2 + 3i)^2$.

Apply the formula: calculate $a^2 = (2)^2$, $2ab = 2 \times 2 \times 3i$, and $b^2 = (3i)^2$ separately.

Remember that $i^2 = -1$, so when you calculate $b^2 = (3i)^2$, rewrite it as $3^2 \times i^2$ and simplify accordingly.

Combine all the terms from the previous step to write the expression in the form $x + yi$, where $x$ and $y$ are real numbers, which is the standard form of a complex number.

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주요 개념

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Complex Numbers

Complex numbers are numbers in the form a + bi, where a and b are real numbers and i is the imaginary unit with the property i² = -1. Understanding how to work with complex numbers is essential for performing operations like addition, multiplication, and exponentiation.

Binomial Expansion

Binomial expansion involves expanding expressions raised to a power, such as (a + b)², using the formula (a + b)² = a² + 2ab + b². This technique helps simplify powers of binomials, including those with complex terms.

Standard Form of a Complex Number

The standard form of a complex number is expressed as a + bi, where a is the real part and b is the imaginary part. After performing operations, rewriting the result in this form clearly separates the real and imaginary components.

Find each product and write the result in standard form. (2 + 3i)2