In Exercises 9–20, find each product and write the result in standard form.

(3 + 5i)(3 − 5i)

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Recognize that the expression is a product of two complex conjugates: $(3 + 5i)$ and $(3 - 5i)$.

Recall the formula for the product of conjugates: $(a + bi)(a - bi) = a^2 + b^2$, where $a$ and $b$ are real numbers.

Identify $a = 3$ and $b = 5$ from the given expression.

Calculate $a^2$ and $b^2$ separately: \$3^2$ and $5^2$.

Add the results from the previous step to write the product in standard form: $a^2 + b^2$.

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주요 개념

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

Complex Numbers and Standard Form

Complex numbers are expressed in the form a + bi, where a is the real part and b is the imaginary part. Writing a complex number in standard form means expressing it explicitly as a sum of its real and imaginary components.

Multiplication of Complex Numbers

To multiply complex numbers, use the distributive property (FOIL method), multiplying each term in the first complex number by each term in the second. Remember that i² equals -1, which simplifies the product.

Difference of Squares Formula

The product (a + b)(a - b) equals a² - b². This formula applies to complex conjugates like (3 + 5i)(3 - 5i), simplifying the multiplication by turning it into a difference of squares involving real numbers and imaginary parts.

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