Textbook Question
Identify each number as real, complex, pure imaginary, or nonreal com-plex. (More than one of these descriptions will apply.) √-36
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1. Equations & Inequalities
Complex Numbers
1:30 minutesProblem 25
Textbook Question
Perform each operation. Write answers in standard form. (5-11i)(5+11i)
Verified step by step guidance1
Recognize that the expression is a product of two complex conjugates: \((5 - 11i)(5 + 11i)\).
Recall the formula for the product of conjugates: \((a - bi)(a + bi) = a^2 + b^2\), where \(a = 5\) and \(b = 11\).
Calculate \(a^2\) which is \$5^2\( and \)b^2\( which is \)11^2$ separately.
Add the results from the previous step to get \(a^2 + b^2\).
Write the final answer in standard form \(x + yi\), noting that the imaginary part will be zero because the product of conjugates is a real number.
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Video duration:
1mKey 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 answer explicitly as a sum of a real number and an imaginary number. Understanding this form is essential for correctly presenting the result of operations involving complex numbers.
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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, remembering that i² = -1. This process combines like terms to simplify the expression.
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Simplifying Using i² = -1
Since i is the imaginary unit, i² equals -1. This identity allows you to convert powers of i into real numbers, simplifying expressions involving complex numbers. Applying this rule is crucial when combining terms after multiplication.
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