(3i^(5)+2i^(7)+i^(9))/(i^(6)+2i^(8)+3i^(18))

Jasonbradley

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Jun 16, 2024

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Simplifying Complex Expressions: A Step-by-Step Guide

This article will guide you through the process of simplifying the complex expression:

(3i^(5)+2i^(7)+i^(9))/(i^(6)+2i^(8)+3i^(18))

Let's break down the steps involved:

Understanding Complex Numbers and Powers

Complex numbers are numbers of the form a + bi, where a and b are real numbers and i is the imaginary unit, defined as the square root of -1.

Powers of the imaginary unit follow a cyclical pattern:

• i^1 = i
• i^2 = -1
• i^3 = -i
• i^4 = 1

This pattern repeats for higher powers. To determine the value of a power of i, divide the exponent by 4 and look at the remainder:

• Remainder 0: i^4 = 1
• Remainder 1: i^1 = i
• Remainder 2: i^2 = -1
• Remainder 3: i^3 = -i

Simplifying the Expression

Let's apply this knowledge to simplify our expression:

1. Simplify the powers of i:

   • i^5 = i^4 * i = 1 * i = i
   • i^7 = i^4 * i^3 = 1 * -i = -i
   • i^9 = i^4 * i^5 = 1 * i = i
   • i^6 = i^4 * i^2 = 1 * -1 = -1
   • i^8 = i^4 * i^4 = 1 * 1 = 1
   • i^18 = i^4 * i^4 * i^4 * i^4 * i^2 = 1 * 1 * 1 * 1 * -1 = -1

2. Substitute the simplified powers:

   (3i + 2(-i) + i) / (-1 + 2(1) + 3(-1))

3. Combine like terms:

   (2i) / (-2)

4. Simplify:

   -i

Therefore, the simplified form of the expression (3i^(5)+2i^(7)+i^(9))/(i^(6)+2i^(8)+3i^(18)) is -i.