Simplify Each of the Following Powers of i

Oct 13, 2025 by leolavanderia.com.br 43 views

Simplify Each of the Following Powers of i

The imaginary unit, denoted by $i$ , is a fundamental concept in mathematics, particularly in algebra and geometry. It is defined as the square root of $-1$ , denoted by $i = \sqrt{-1}$ . The powers of $i$ are used extensively in various mathematical operations, including complex numbers, trigonometry, and calculus. In this article, we will simplify each of the following powers of $i$ .

The powers of $i$ follow a cyclical pattern, repeating every four powers. To simplify each power of $i$ , we need to understand this pattern.

$i^0 = 1$ $i^1 = i$ $i^2 = -1$ $i^3 = -i$ $i^4 = 1$ $i^5 = i$ $i^6 = -1$ $i^7 = -i$ $i^8 = 1$ $i^9 = i$ $i^{10} = -1$ $i^{11} = -i$ $i^{12} = 1$ $i^{13} = i$ $i^{14} = -1$ $i^{15} = \boxed{-i}$

To simplify higher powers of $i$ , we can use the following properties:

$i^4 = 1$
$i^5 = i$
$i^6 = -1$
$i^7 = -i$

Using these properties, we can simplify higher powers of $i$ by reducing them to one of these four powers.

Example 1

Simplify $i^{15}$ .

Using the property $i^4 = 1$ , we can rewrite $i^{15}$ as $(i^4)^3 \cdot i^3$ . Since $i^4 = 1$ , we have $(i^4)^3 = 1^3 = 1$ . Therefore, $i^{15} = 1 \cdot i^3 = -i$ .

Example 2

Simplify $i^{27}$ .

Using the property $i^4 = 1$ , we can rewrite $i^{27}$ as $(i^4)^6 \cdot i^3$ . Since $i^4 = 1$ , we have $(i^4)^6 = 1^6 = 1$ . Therefore, $i^{27} = 1 \cdot i^3 = -i$ .

In conclusion, the powers of $i$ follow a cyclical pattern, repeating every four powers. To simplify each power of $i$ , we need to understand this pattern and use the properties of $i^4 = 1$ , $i^5 = i$ , $i^6 = -1$ , and $i^7 = -i$ . By applying these properties, we can simplify higher powers of $i$ and reduce them to one of these four powers.

The powers of $i$ have numerous applications in mathematics, particularly in complex numbers, trigonometry, and calculus. Some of the key applications include:

Complex Numbers: The powers of $i$ are used to represent complex numbers in the form $a + bi$ , where $a$ and $b$ are real numbers.
Trigonometry: The powers of $i$ are used to represent trigonometric functions, such as sine and cosine, in terms of complex numbers.
Calculus: The powers of $i$ are used to represent complex functions, such as the exponential function, in terms of complex numbers.

Q: What is the imaginary unit i?

A: The imaginary unit, denoted by $i$ , is a fundamental concept in mathematics, particularly in algebra and geometry. It is defined as the square root of $-1$ , denoted by $i = \sqrt{-1}$ .

Q: What are the powers of i?

A: The powers of $i$ follow a cyclical pattern, repeating every four powers. The powers of $i$ are:

$i^0 = 1$ $i^1 = i$ $i^2 = -1$ $i^3 = -i$ $i^4 = 1$ $i^5 = i$ $i^6 = -1$ $i^7 = -i$ $i^8 = 1$ $i^9 = i$ $i^{10} = -1$ $i^{11} = -i$ $i^{12} = 1$ $i^{13} = i$ $i^{14} = -1$ $i^{15} = -i$

Q: How do I simplify higher powers of i?

A: To simplify higher powers of $i$ , we can use the following properties:

$i^4 = 1$
$i^5 = i$
$i^6 = -1$
$i^7 = -i$

Using these properties, we can simplify higher powers of $i$ by reducing them to one of these four powers.

Q: What are some examples of simplifying higher powers of i?

A: Here are some examples of simplifying higher powers of $i$ :

Simplify $i^{15}$ . Using the property $i^4 = 1$ , we can rewrite $i^{15}$ as $(i^4)^3 \cdot i^3$ . Since $i^4 = 1$ , we have $(i^4)^3 = 1^3 = 1$ . Therefore, $i^{15} = 1 \cdot i^3 = -i$ .
Simplify $i^{27}$ . Using the property $i^4 = 1$ , we can rewrite $i^{27}$ as $(i^4)^6 \cdot i^3$ . Since $i^4 = 1$ , we have $(i^4)^6 = 1^6 = 1$ . Therefore, $i^{27} = 1 \cdot i^3 = -i$ .

Q: What are some applications of powers of i?

A: The powers of $i$ have numerous applications in mathematics, particularly in complex numbers, trigonometry, and calculus. Some of the key applications include:

Complex Numbers: The powers of $i$ are used to represent complex numbers in the form $a + bi$ , where $a$ and $b$ are real numbers.
Trigonometry: The powers of $i$ are used to represent trigonometric functions, such as sine and cosine, in terms of complex numbers.
Calculus: The powers of $i$ are used to represent complex functions, such as the exponential function, in terms of complex numbers.

Q: Why are powers of i important in mathematics?

A: The powers of $i$ are important in mathematics because they provide a way to represent complex numbers and trigonometric functions in a compact and elegant way. They also play a crucial role in various mathematical operations, such as complex analysis and calculus.

Q: Can I use powers of i in real-world applications?

A: Yes, powers of $i$ have numerous real-world applications, including:

Signal Processing: Powers of $i$ are used in signal processing to represent complex signals and filter out noise.
Control Systems: Powers of $i$ are used in control systems to represent complex systems and design control algorithms.
Electrical Engineering: Powers of $i$ are used in electrical engineering to represent complex electrical circuits and design electronic circuits.

Q: How can I learn more about powers of i?

A: There are many resources available to learn more about powers of $i$ , including:

Textbooks: There are many textbooks available that cover the topic of powers of $i$ in detail.
Online Courses: There are many online courses available that cover the topic of powers of $i$ .
Research Papers: There are many research papers available that cover the topic of powers of $i$ in detail.

In conclusion, powers of $i$ are a fundamental concept in mathematics, particularly in algebra and geometry. By understanding the cyclical pattern of the powers of $i$ and using the properties of $i^4 = 1$ , $i^5 = i$ , $i^6 = -1$ , and $i^7 = -i$ , we can simplify higher powers of $i$ and reduce them to one of these four powers. The applications of powers of $i$ are numerous and far-reaching, and they play a crucial role in various mathematical operations.