Find each product and write the result in standard form. (3 + 5i)(3 - 5i)

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

Recall the formula for the product of conjugates: $(a + bi)(a - bi) = a^2 + b^2$, where $a = 3$ and $b = 5$.

Square the real part: calculate $3^2$.

Square the imaginary coefficient: calculate $5^2$.

Add the two results together to write the product in standard form $a + bi$, noting that the imaginary parts cancel out.

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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. They extend the real number system and are used to represent quantities that have both real and imaginary parts.

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Multiplication of Complex Numbers

To multiply complex numbers, use the distributive property (FOIL method) and apply the rule i² = -1 to simplify. Multiply each term in the first complex number by each term in the second, then combine like terms to get the product.

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Standard Form of a Complex Number

The standard form of a complex number is a + bi, where a is the real part and b is the coefficient of the imaginary part. After multiplication, simplify the expression to this form for clarity and consistency.

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