Question

Find the value of i15+2 i109+ i100- i17.

Answer 1

Step-by-step explanation:

Explanation:

Remember that

i

2

=

−

1

.

Thus,

i

4

=

(

i

2

)

2

=

(

−

1

)

2

=

1

Also, remember the power rule

a

m

⋅

a

n

=

a

m

+

n

Thus, you have

i

15

=

i

4

+

4

+

4

+

3

=

i

4

⋅

i

4

⋅

i

4

⋅

i

3

=

1

⋅

1

⋅

1

⋅

i

3

=

i

3

=

i

2

⋅

i

=

−

1

⋅

i

=

−

i

===========

Also, I'd like to offer you a more general solution for

i

n

, with

n

being any positive integer.

Try to recognize the pattern:

i

=

i

i

2

=

−

1

i

3

=

i

2

⋅

i

=

−

1

⋅

i

=

−

i

i

4

=

i

3

⋅

i

=

−

i

⋅

i

=

−

i

2

=

1

i

5

=

i

4

⋅

i

=

1

⋅

i

=

i

i

6

=

i

4

⋅

i

2

=

−

1

...

So, basically, the power of

i

is always

i

,

−

1

,

−

i

,

1

, and then repeat.

Thus, to compute

i

n

, there are four possibilites:

if

n

can be divided by

4

, then

i

n

=

1

if

n

can be divided by

2

(but not by

4

), then

i

n

=

−

1

if

n

is an odd number but

n

−

1

can be divided by

4

, then

i

n

=

i

if

n

is an odd number but

n

+

1

can be divided by

4

, then

i

n

=

−

i

Described in a more formal way,

i

n

=

⎧

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎩

1

n

=

4

k

i

n

=

4

k

+

1

−

1

n

=

4

k

+

2

−

i

n

=

4

k

+

3

for

k

∈

N

0

.