Find each product. Write answers in standard form. i(3-4i)(3+4i)

검증된 단계별 안내

First, recognize that the expression is a product of three factors: $i$, $(3 - 4i)$, and $(3 + 4i)$.

Next, multiply the two complex conjugates $(3 - 4i)$ and $(3 + 4i)$ using the difference of squares formula: $(a - bi)(a + bi) = a^2 + b^2$. So, calculate $3^2 + 4^2$.

After finding the product of the conjugates, multiply the result by $i$.

Recall that $i^2 = -1$, and use this fact to simplify any powers of $i$ that appear during multiplication.

Finally, write the resulting expression in standard form $a + bi$, where $a$ and $b$ are real numbers.

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주요 개념

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

Complex Numbers and Imaginary Unit

Complex numbers have 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 working with complex expressions.

Multiplication of Complex Numbers

Multiplying complex numbers involves using the distributive property (FOIL method) and combining like terms, especially handling i² terms by replacing them with -1. This process helps simplify products of complex expressions.

Standard Form of Complex Numbers

The standard form of a complex number is a + bi, where a and b are real numbers. After multiplication, the result should be simplified and written in this form, separating the real and imaginary parts clearly.