Find each product and write the result in standard form. - 3i(7i - 5)

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Distribute the term $ 3i $ to each term inside the parentheses $ (7i - 5) $. This means multiplying $ 3i $ by $ 7i $ and $ 3i $ by $ -5 $.

When multiplying $ 3i $ by $ 7i $, use the property of exponents: $ i \cdot i = i^2 $. So, $ 3i \cdot 7i = 21i^2 $.

When multiplying $ 3i $ by $ -5 $, treat $ i $ as a variable and multiply the constants: $ 3i \cdot -5 = -15i $.

Combine the results from the distribution: $ 21i^2 - 15i $.

Recall that $ i^2 = -1 $ (a fundamental property of imaginary numbers). Substitute $ -1 $ for $ i^2 $ in $ 21i^2 $, and simplify the expression to write it 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

Complex numbers are numbers that have a real part and an imaginary part, expressed in the form a + bi, where 'a' is the real part and 'b' is the coefficient of the imaginary unit 'i', which is defined as the square root of -1. Understanding complex numbers is essential for performing operations involving 'i', especially in multiplication and addition.

Distributive Property

The distributive property states that a(b + c) = ab + ac, allowing us to multiply a single term by a sum or difference. This property is crucial when expanding expressions, such as multiplying a complex number by a binomial, as it ensures that each term is correctly multiplied and combined.

Standard Form of Complex Numbers

The standard form of a complex number is expressed as a + bi, where 'a' and 'b' are real numbers. When performing operations with complex numbers, it is important to simplify the result into this form, ensuring that the real and imaginary parts are clearly separated for easier interpretation and further calculations.

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