Simplifying Complex Expressions: (82i^4)+(37i^8)(4+i^9)
This article will walk through the process of simplifying the complex expression: (82i^4)+(37i^8)(4+i^9).
Understanding the Properties of Imaginary Numbers
Before we can simplify the expression, we need to recall some key properties of imaginary numbers:
 i is defined as the square root of 1.
 i^2 = 1
 i^3 = i^2 * i = 1 * i = i
 i^4 = i^2 * i^2 = (1) * (1) = 1
 This pattern of i^n repeats every four powers.
Simplifying the Expression

Simplify the powers of i:
 i^4 = 1
 i^8 = (i^4)^2 = 1^2 = 1
 i^9 = i^4 * i^5 = 1 * i * (i^4) = i

Substitute the simplified values:
 (8  2 * 1) + (3  7 * 1)  (4 + i)

Simplify the real and imaginary components:
 (8  2 + 3  7  4) + (1)i

Combine the terms:
 0  i
Conclusion
Therefore, the simplified form of the complex expression (82i^4)+(37i^8)(4+i^9) is i.