The powers of i ( i^{n }) are proven within the desk under and these might be computed fairly simply when n > 0 and n < 0.

In case you didn’t fairly perceive the data within the desk, preserve studying to see the logic behind it!

## Powers of i when n > 0

i^{1} = i

i^{2 } = – 1

i^{3} = i^{2} × i = -i

i^{4} = i^{2} × i^{2} = 1

i^{5} = i^{4} × i = i

i^{6} = i^{4} × i^{2} = -1

i^{7} = i^{4} × i^{3} = -i

i^{8} = i^{4} × i^{4} = 1

Discover the sample i, -1, -i, 1, … repeats after the primary 4 advanced numbers. Normally, if n is an integer greater than zero, the worth of i^{n} might be discovered by dividing n by 4 and inspecting the rest.

Did you make the next observations concerning the powers of i?

- For i
^{4}and that i^{8}, the rest is 0 after we divide 4 and eight by 4. - For i
^{3}and that i^{7}, the rest is 3 after we divide 3 and seven by 4. - For i
^{2}and that i^{6}, the rest is 2 after we divide 2 and 6 by 4. - For i
^{1}and that i^{5}, the rest is 1 after we divide 1 and 5 by 4.

Conclusion

Let n > 0 and R is the rest when n is split by 4

If R = 1, i^{n} = i

If R = 2, i^{n} = -1

If R = 3, i^{n} = -i

If R = 0, i^{n} = 1

## Powers of i when n < 0

i^{-1} = 1 / i = (1 × i) / (i × i) = i / i^{2} = i / -1 = -i

i^{-2 }= 1 / i^{2} = 1 / -1 = -1

i^{-3} = 1 / i^{3} = 1 / -i = (1 × i) / (-i × i) = i / 1 = i

i^{-4} = 1 / i^{4} = 1 / 1 = 1

i^{-5} = 1 / i^{5} = 1 / i = -i

i^{-6} = 1 / i^{6} = 1 / -1 = -1

i^{-7} = 1 / i^{7} = 1 / -i = i

i^{-8} = 1 / i^{8} = 1 / 1 = 1

i^{-1 }=-i

i^{-2 }= -1

i^{-3} = i

i^{-4} = 1

i^{-5 }=-i

i^{-6 }= -1

i^{-7} = i

i^{-8} = 1

Discover the sample -i, -1, i, 1, … repeats after the primary 4 advanced numbers. Normally, if n is an integer smaller than zero, the worth of i^{n} might be discovered by dividing n by 4 and inspecting the rest.

Did you make the next observations concerning the powers of i?

- For i
^{-4}and that i^{-8}, the rest is 0 after we divide 4 and eight by 4. - For i
^{-3}and that i^{-7}, the rest is -3 after we divide -3 and -7 by 4. - For i
^{-2}and that i^{-6}, the rest is -2 after we divide -2 and -6 by 4. - For i
^{-1}and that i^{-5}, the rest is -1 after we divide 1 and 5 by 4.

Conclusion

Let n < 0 and R is the rest when n is split by 4

If R = -1, i^{n} = -i

If R = -2, i^{n} = -1

If R = -3, i^{n} = i

If R = 0, i^{n} = 1

## A couple of examples displaying the best way to discover the powers of i.

**Instance #1:**

i^{67}

67 divided 4 offers a the rest of three. Since n is constructive, i^{67} = -i

**Instance #2:**

i^{-67}

-67 divided 4 offers a the rest of -3. Since n is detrimental, i^{-67} = i

**Instance #3:**

i^{36}

36 divided 4 offers a the rest of 0. Since n is constructive, i^{36} = 1

**Instance #4:**

i^{-36}

-36 divided 4 offers a the rest of 0. Since n is detrimental, i^{-36} = 1