Finding The Value Of I^7 A Step-by-Step Guide

In the realm of mathematics, particularly within complex numbers, the imaginary unit 'i' plays a pivotal role. It's defined as the square root of -1, a concept that extends the real number system to encompass numbers with imaginary components. This exploration into complex numbers opens up a vast landscape of mathematical possibilities, enabling us to solve equations and understand phenomena that are beyond the scope of real numbers alone. Mastering the manipulation of 'i' and its powers is crucial for anyone venturing into areas like electrical engineering, quantum mechanics, and advanced calculus. In this comprehensive guide, we will embark on a journey to unravel the mystery behind finding the value of i raised to the power of 7 (i^7). We'll dissect the fundamental properties of i, explore the cyclic nature of its powers, and ultimately arrive at the solution in a clear and concise manner. We'll not only focus on arriving at the correct answer but also aim to build a solid understanding of the underlying principles. This approach will empower you to confidently tackle similar problems and appreciate the elegance of complex number theory.

At the heart of complex numbers lies the imaginary unit, denoted by the symbol 'i'. Unlike real numbers, which can be plotted on a number line, i represents a number whose square is -1. This seemingly simple definition unlocks a world of mathematical possibilities. It allows us to deal with square roots of negative numbers, which are undefined within the real number system. The fundamental definition of i is:

i = √(-1)

This single equation is the cornerstone of complex number theory. From it, we can derive a series of interesting properties and relationships. Squaring both sides of the equation, we get:

i^2 = (√(-1))^2 = -1

This is the second crucial identity to remember. i squared equals -1. This seemingly small fact has profound implications for how we manipulate and understand complex numbers. It forms the basis for simplifying expressions involving powers of i, and it allows us to express complex numbers in the standard form a + bi, where a and b are real numbers. This representation is essential for performing arithmetic operations on complex numbers, such as addition, subtraction, multiplication, and division.

Understanding the definition of i and the value of i^2 is paramount for solving problems involving complex numbers. It is the key to unraveling the cyclic nature of powers of i, which we will explore in detail in the following sections. Grasping these fundamental concepts will not only enable you to find the value of i^7 but will also provide a solid foundation for further exploration into the fascinating world of complex numbers.

The beauty of working with the imaginary unit i lies in the cyclic pattern that emerges when we consider its powers. This cyclical nature allows us to simplify higher powers of i into one of four basic values: i, -1, -i, and 1. This cyclical pattern is a direct consequence of the definition of i and its square. Let's delve into this pattern to understand how it works:

• i^1 = i (By definition)
• i^2 = -1 (As we established earlier)
• i^3 = i^2 * i = -1 * i = -i
• i^4 = i^2 * i^2 = (-1) * (-1) = 1

Notice what happens when we reach i^4. It equals 1. This is a crucial observation. Because i^4 = 1, any power of i that is a multiple of 4 will also equal 1. For example, i^8 = (i⁴⁾2 = 1^2 = 1, and so on. This cyclical pattern repeats itself every four powers:

• i^5 = i^4 * i = 1 * i = i
• i^6 = i^4 * i^2 = 1 * (-1) = -1
• i^7 = i^4 * i^3 = 1 * (-i) = -i
• i^8 = i^4 * i^4 = 1 * 1 = 1

and so on. This pattern (i, -1, -i, 1) continues indefinitely. This cyclical nature simplifies the process of finding the value of any power of i. Instead of calculating each power individually, we can use the fact that the pattern repeats every four powers to reduce the exponent to a smaller, more manageable value. This cyclical behavior is not just a mathematical curiosity; it is a fundamental property of complex numbers that is used extensively in various applications.

To find the value of any power of i, we can divide the exponent by 4 and consider the remainder. The remainder will tell us where we are in the cycle. This is the key to solving problems like finding the value of i^7. By understanding the cyclical nature of powers of i, we can efficiently determine the value of any power of i without having to perform lengthy calculations. This approach not only saves time but also provides a deeper understanding of the underlying principles of complex number theory.

Now that we have a firm grasp on the cyclic nature of powers of i, we can confidently tackle the problem of finding the value of i^7. The key is to utilize the cyclical pattern we identified earlier: i, -1, -i, 1. To find i^7, we need to determine where 7 falls within this cycle. We can do this by dividing the exponent 7 by 4:

7 ÷ 4 = 1 remainder 3

The quotient (1) tells us that the cycle repeats once completely. The remainder (3) is the crucial piece of information. It indicates that i^7 is equivalent to the third power in the cycle. In other words,

i^7 = i^(41 + 3) = (i⁴⁾1 * i^3*

Since we know that i^4 = 1, we can simplify this expression:

i^7 = 1^1 * i^3 = i^3

Now we just need to recall the value of i^3. From our previous exploration of the cyclical pattern, we know that:

i^3 = i^2 * i = -1 * i = -i

Therefore,

i^7 = -i

This straightforward calculation demonstrates the power of understanding the cyclical nature of powers of i. By dividing the exponent by 4 and focusing on the remainder, we can quickly and efficiently determine the value of any power of i. This method is not only applicable to i^7 but can be generalized to solve for any i^n, where n is an integer. The elegance of this approach lies in its simplicity and its reliance on a fundamental property of complex numbers. Mastering this technique is essential for anyone working with complex numbers, as it streamlines calculations and provides a deeper understanding of the underlying principles.

In conclusion, we have successfully navigated the process of finding the value of i^7 by leveraging the cyclical nature of powers of the imaginary unit i. We started by defining i as the square root of -1 and establishing the crucial identity i^2 = -1. We then explored the repeating pattern of powers of i: i, -1, -i, 1, which forms the cornerstone of simplifying expressions involving imaginary units. By dividing the exponent 7 by 4 and focusing on the remainder, we efficiently determined that i^7 = -i.

This exercise not only provides a specific solution but also underscores the importance of understanding the fundamental properties of complex numbers. The ability to manipulate powers of i is a crucial skill in various areas of mathematics, physics, and engineering. Complex numbers are used extensively in electrical engineering to analyze alternating current circuits, in quantum mechanics to describe the behavior of particles, and in signal processing to manipulate and analyze signals.

The cyclical nature of powers of i is not just a mathematical curiosity; it is a fundamental property that simplifies calculations and provides a deeper understanding of complex number behavior. By mastering this concept, you equip yourself with a powerful tool for tackling more complex problems involving imaginary and complex numbers. The journey of understanding i^7 serves as a stepping stone to exploring the rich and fascinating world of complex number theory, opening doors to advanced mathematical concepts and real-world applications.

Therefore, the solution to the problem “Find the value of i^7” is (A) -i.