Textbook Question
Find each product. Write answers in standard form. (3-i)(3+1)(2-6i)
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1. Equations & Inequalities
Complex Numbers
1:08 minutesProblem 90
Textbook Question
Simplify each power of i. i29
Verified step by step guidance1
Recall that the powers of the imaginary unit \(i\) cycle every 4 powers: \(i^1 = i\), \(i^2 = -1\), \(i^3 = -i\), and \(i^4 = 1\).
To simplify \(i^{29}\), find the remainder when 29 is divided by 4, since the powers repeat every 4.
Calculate \(29 \div 4\) which gives a quotient and a remainder. The remainder determines the equivalent power of \(i\).
Express \(i^{29}\) as \(i^{4k + r}\) where \(k\) is the quotient and \(r\) is the remainder from the division.
Use the fact that \(i^{4k} = (i^4)^k = 1^k = 1\), so \(i^{29} = i^r\), and then substitute the value of \(r\) to find the simplified form.
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1mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Imaginary Unit i
The imaginary unit i is defined as the square root of -1, satisfying i² = -1. It is the fundamental building block of complex numbers and is used to extend the real number system to include solutions to equations like x² + 1 = 0.
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Powers of i
Powers of i and Their Cyclic Pattern
Powers of i repeat in a cycle of four: i¹ = i, i² = -1, i³ = -i, and i⁴ = 1. This pattern repeats for higher powers, so simplifying i raised to any integer power involves finding the remainder when the exponent is divided by 4.
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Modular Arithmetic for Exponent Simplification
Modular arithmetic helps simplify powers by reducing the exponent modulo 4 in this context. For example, to simplify i²⁹, compute 29 mod 4 = 1, so i²⁹ = i¹ = i. This technique streamlines calculations involving cyclic patterns.
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5:17Arithmetic Sequences - General Formula
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