https://file.aiquickdraw.com/mathbot/file/1729022435510w3nog1nh.jpg

Let's simplify the given powers of the imaginary unit \(i\). Remember that the imaginary unit \(i\) follows this cyclical pattern:

\[
i^1 = i, \quad i^2 = -1, \quad i^3 = -i, \quad i^4 = 1
\]
This pattern repeats every 4 powers. Therefore, to simplify any power of \(i\), we divide the exponent by 4 and check the remainder.

### a. \(i^{48}\)
\[
48 \div 4 = 12 \text{ remainder } 0
\]
Since the remainder is 0, we use \(i^4 = 1\):
\[
i^{48} = 1
\]

### b. \(i^{23}\)
\[
23 \div 4 = 5 \text{ remainder } 3
\]
Since the remainder is 3, we use \(i^3 = -i\):
\[
i^{23} = -i
\]

### c. \(i^{50}\)
\[
50 \div 4 = 12 \text{ remainder } 2
\]
Since the remainder is 2, we use \(i^2 = -1\):
\[
i^{50} = -1
\]

### d. \(i^{-19}\)
For negative exponents, we can first simplify the positive exponent. So, let's simplify \(i^{19}\):
\[
19 \div 4 = 4 \text{ remainder } 3
\]
Since the remainder is 3, \(i^{19} = -i\). Now, since we had a negative exponent (\(i^{-19}\)), the result is the reciprocal of \(-i\), which is still \(-i\) (because dividing by an imaginary number like \(i\) maintains the sign).

Thus:
\[
i^{-19} = -i
\]

### Final answers:
a. \(i^{48} = 1\)  
b. \(i^{23} = -i\)  
c. \(i^{50} = -1\)  
d. \(i^{-19} = -i\)

Would you like further clarification on any part of this solution? Here are five related questions you can explore:

1. How do powers of \(i\) repeat in cycles?
2. How do we deal with negative exponents of imaginary numbers?
3. Can the cyclic behavior of \(i\) be generalized to other imaginary units in complex numbers?
4. Why does \(i^4 = 1\) and what does this tell us about the nature of \(i\)?
5. What happens when you raise \(i\) to non-integer exponents?

**Tip:** When working with powers of \(i\), always reduce the exponent modulo 4 to simplify the expression.

Simplify the powers of the imaginary unit i: a. i^48, b. i^23, c. i^50, d. i^-19

Learn how to simplify powers of the imaginary unit i, including i^48, i^23, i^50, and i^-19. Understand the cyclical nature of powers of i and apply basic rules of exponents and imaginary numbers. Suitable for Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation