The Value of Iota Raised to the Power of 7

In the realm of complex numbers, the value of $i$i (the imaginary unit) is fundamental. By definition, $i$i is the square root of -1, or $i^2=-1$i2=−1. When dealing with powers of $i$i, it's essential to understand the cyclical nature of these powers.

To find the value of $i^7$i7, we can utilize the fact that the powers of $i$i repeat in a cycle of four. Specifically, the cycle is as follows:

• $i^1=i$i1=i
• $i^2=-1$i2=−1
• $i^3=-i$i3=−i
• $i^4=1$i4=1

After $i^4$i4, the cycle repeats. This cyclical pattern can be used to simplify expressions involving higher powers of $i$i.

For $i^7$i7, we first recognize that 7 can be broken down using the cycle length of 4. We can compute this as:

$7÷4=1\text{ remainder }3$7÷4=1 remainder 3

This tells us that $i^7$i7 corresponds to $i^3$i3 within the cycle. From our cycle list, $i^3=-i$i3=−i. Therefore:

$i^7=-i$i7=−i

To further illustrate this, let’s break it down with a quick example:

1. Start with the known powers:

   • $i^1=i$i1=i
   • $i^2=-1$i2=−1
   • $i^3=-i$i3=−i
   • $i^4=1$i4=1

2. Since the powers repeat every four, $i^5=i$i5=i, $i^6=-1$i6=−1, and $i^7=-i$i7=−i, as per our calculation.

Thus, the value of $i$i raised to the power of 7 is $-i$−i.

Summary: The calculation and cyclic nature of the powers of $i$i allow us to determine that $i^7$i7 simplifies to $-i$−i. Understanding this cyclical pattern is crucial for simplifying complex number computations.