Chose a power of
i
(with an exponent 13 or higher) that is equivalent to : i^3
3
. Explain how you arrived at your answer.
To find a power of i that is equivalent to i^3, we can start by calculating the value of i^3 first.
Using the formula for powers of i, where i^2 = -1, we can simplify i^3 as follows:
i^3 = (i^2) * i
= (-1) * i
= -i
Now, we need to find a power of i with an exponent 13 or higher that is equivalent to -i.
If we multiply i by itself 4 times, we get:
i^4 = (i^2)^2
= (-1)^2
= 1
Therefore, i^4 is equal to 1.
Since i^3 = -i, we can simplify i^3 * i = -i * i = -i^2.
Using the value of i^4 = 1, we can substitute -i^2 as follows:
-i^2 = -1 * (i^2)
= -1 * 1
= -1
Therefore, i^3 * i = -1.
Hence, a power of i (with an exponent 13 or higher) that is equivalent to i^3 is -1.
To find a power of i that is equivalent to i^3, we can start by finding the pattern of powers of i.
The imaginary unit i is defined as the square root of -1. So, i^2 = -1.
Now, let's calculate the powers of i to observe the pattern:
i^1 = i
i^2 = -1
i^3 = -1 × i = -i
i^4 = (-1) × (-1) = 1
i^5 = 1 × i = i
i^6 = i × i = -1
i^7 = -1 × i = -i
i^8 = (-1) × (-1) = 1
...
We can see that the pattern repeats every 4 powers of i. Since i^3 equals -i, we can calculate the equivalent power of i by subtracting 3 from any multiple of 4.
To find a power of i (with an exponent 13 or higher) that is equivalent to i^3, we can subtract 3 from 13:
i^13 = i^(12 + 1) = (i^4)^3 × i = (1)^3 × i = i
So, i^13 is equivalent to i.