Textbook Question
In Exercises 21–28, divide and express the result in standard form. 2/(3 - i)
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1. Equations & Inequalities
The Imaginary Unit
3:43 minutesProblem 41
Textbook Question
In Exercises 37–52, perform the indicated operations and write the result in standard form. (- 2 + √-4)2
Verified step by step guidance1
Step 1: Recognize that the expression involves a square of a binomial, (-2 + √-4)^2. To simplify, use the formula for the square of a binomial: (a + b)^2 = a^2 + 2ab + b^2.
Step 2: Identify the terms in the binomial. Here, a = -2 and b = √-4. Note that √-4 involves the imaginary unit i, since the square root of a negative number is not defined in the real numbers. Rewrite √-4 as 2i, where i = √-1.
Step 3: Substitute the values of a and b into the formula. The expression becomes (-2)^2 + 2(-2)(2i) + (2i)^2.
Step 4: Simplify each term. (-2)^2 = 4, 2(-2)(2i) = -8i, and (2i)^2 = 4i^2. Recall that i^2 = -1, so 4i^2 = 4(-1) = -4.
Step 5: Combine the simplified terms. Add the real parts (4 and -4) and the imaginary part (-8i) to write the result in standard form, which is a + bi, where a is the real part and b is the coefficient of the imaginary part.
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3mKey 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 square roots of negative numbers, as seen in this problem.
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Standard Form of Complex Numbers
The standard form of a complex number is a + bi, where 'a' and 'b' are real numbers. When performing operations on complex numbers, it is important to express the final result in this form to clearly identify the real and imaginary components. This helps in further calculations and applications in various fields, including engineering and physics.
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Exponentiation of Complex Numbers
Exponentiation of complex numbers involves raising a complex number to a power, which can be done using the distributive property and the properties of exponents. In this case, squaring the complex number requires applying the formula (a + bi)² = a² + 2abi + (bi)², where (bi)² equals -b². This process is crucial for simplifying the expression and obtaining the result in standard form.
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