Step-by-step explanation
1. Find the highest multiple of 4 that is less than or equal to the exponent of i
When i is raised to increasing powers, its values will begin repeating themselves every four terms indefinitely:
and so on.
The results start repeating after , which is a pattern that continues every four terms forever. We can use this pattern to figure out i raised to any power.
Divide the power of the i (463) by :
Multiply 4 by 115:
460 is the highest multiple of 4 that is less than or equal to 463.
2. Calculate the power of i
Expand the power using the rule:
Rewrite 460 as a multiple of 4:
Expand the power using the rule:
Because :
Because 1 raised to any power equals 1:
Simplify according to the pattern of the powers of i:
, , ,
The power of equals
How did we do?
Please leave us feedback.Why learn this
Despite their misleading name, imaginary numbers - almost always written as i - are not exactly "imaginary". They were originally described as "imaginary" as an insult because they represent an abstract concept that, when first discovered, did not seem to be particularly useful. They became more widely used and accepted over time, but by that point it was too late! The name stuck. Today, imaginary numbers are frequently used in scientific contexts, such as understanding the behavior of soundwaves, concepts in quantum mechanics, and relativity.
Because imaginary numbers represent the solutions to the square roots of negative numbers, we can use them to solve quadratic equations that have no real roots (meaning they do not intercept the x-axis when graphed).