Textbook Question
Find each product. Write answers in standard form. 3i(2-i)²
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1. Equations & Inequalities
The Imaginary Unit
2:19 minutesProblem 50
Textbook Question
Perform the indicated operations and write the result in standard form. (7-i)(2+3i)
Verified step by step guidance1
Identify the problem as multiplying two complex numbers: \((7 - i)\) and \((2 + 3i)\).
Use the distributive property (FOIL method) to expand the product: multiply each term in the first complex number by each term in the second complex number.
Calculate each product: \(7 \times 2\), \(7 \times 3i\), \(-i \times 2\), and \(-i \times 3i\).
Combine the real parts and the imaginary parts separately. Remember that \(i^2 = -1\), so simplify \(-i \times 3i\) accordingly.
Write the final expression in standard form \(a + bi\), where \(a\) is the real part and \(b\) is the coefficient of the imaginary part.
Verified video answer for a similar problem:This video solution was recommended by our tutors as helpful for the problem above
Video duration:
2mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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. The standard form means writing the result explicitly as a sum of a real number and an imaginary number, making it easier to interpret and use in further calculations.
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Multiplying Complex Numbers
Multiplication of Complex Numbers
To multiply complex numbers, use the distributive property (FOIL method) to expand the product. Multiply each term in the first complex number by each term in the second, then combine like terms, remembering that i² = -1.
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Simplifying Using i² = -1
Since i is the imaginary unit with the property i² = -1, any occurrence of i² in the expression should be replaced by -1. This simplification converts imaginary squared terms into real numbers, allowing the expression to be written in standard form.
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