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

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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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.