Textbook Question
Find each product or quotient. Simplify the answers. √-13 * √-13
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1. Equations & Inequalities
Complex Numbers
1:20 minutesProblem 33
Textbook Question
Simplify each power of i. i1001
Verified step by step guidance1
Recall that the imaginary unit \( i \) has the property \( i^2 = -1 \). Powers of \( i \) cycle every 4 steps: \( i^1 = i \), \( i^2 = -1 \), \( i^3 = -i \), and \( i^4 = 1 \).
To simplify \( i^{1001} \), find the remainder when 1001 is divided by 4, since the powers of \( i \) repeat every 4.
Calculate \( 1001 \mod 4 \) to determine the equivalent smaller exponent.
Use the remainder from the previous step to rewrite \( i^{1001} \) as one of \( i^0, i^1, i^2, \) or \( i^3 \), which correspond to \( 1, i, -1, \) or \( -i \) respectively.
Express the simplified form of \( i^{1001} \) based on the equivalent power found.
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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 unit used to extend the real number system to complex numbers, allowing for the representation of numbers involving the square roots of negative values.
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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 large exponents by reducing them modulo a certain number—in this case, 4. By calculating the exponent modulo 4, we determine the equivalent power of i within the four-step cycle, making simplification straightforward.
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5:17Arithmetic Sequences - General Formula
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