How to simplify the power of i?

Understand the Problem

The question is asking how to simplify expressions that involve powers of the imaginary unit 'i', which represents the square root of -1. It hints at understanding the cyclical nature of powers of 'i', where the powers repeat every four terms: i, -1, -i, and 1.

Answer

-1

Steps to Solve

Identify the powers of i The powers of the imaginary unit ( i ) cycle through a pattern every four terms:

( i^1 = i )
( i^2 = -1 )
( i^3 = -i )
( i^4 = 1 )

After ( i^4 ), the cycle repeats.

Determine the exponent modulo 4 To simplify ( i^n ), we need to find ( n \mod 4 ). This tells us where in the cycle the power falls.
Evaluate based on the result from the modulo operation Depending on the result from step 2:

If ( n \mod 4 = 0 ), then ( i^n = 1 )
If ( n \mod 4 = 1 ), then ( i^n = i )
If ( n \mod 4 = 2 ), then ( i^n = -1 )
If ( n \mod 4 = 3 ), then ( i^n = -i )

Example calculation For instance, if we want to simplify ( i^{10} ):
First, calculate ( 10 \mod 4 ), which gives us ( 2 ).

Thus, ( i^{10} = -1 ) since ( 10 \mod 4 = 2 ).

The simplified expression for ( i^{10} ) is -1.

More Information

The imaginary unit ( i ) plays a crucial role in complex numbers and helps in solving equations that do not have real solutions. The cyclical behavior of ( i ) makes it easier to work with high powers.

Tips

Forgetting the cyclical pattern of ( i ).
Miscalculating ( n \mod 4 ).
Confusing between the values when referring to the power of ( i ).

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