Question

i^183486 + i^179562 = ? , where i is a complex number​

Answer 1

Answer:

i^(183486) + i^(179562) = (-2)

Step-by-step explanation:

Before solving our Question, we must know that,

a^(m × n) = (a^m)^n or (a^n)^m

Now, we should also know that,

i¹ = i

i² = (-1)

i³ = -i

i⁴ = 1

Now, I gave the above powers of i so that, if we could convert our equation into the powers of i² or i⁴, so that we will get a real answer with no imaginary parts.

So, we have,

i^(183486) + i^(179562)

We know that,

183486 = 2 × 91743

and

179562 = 2 × 89781

So,

i^(183486) + i^(179562)

= i^(2 × 91743) + i^(2 × 89781)

From the above general equation we stated at the beginning we get,

i^(2 × 91743) + i^(2 × 89781)

= (i²)^(91743) + (i²)^(89781)

We know that,

i² = (-1)

also,

(-1)^(odd number) = (-1)

(-1)^(even number) = 1

So,

(-1)^(91743) + (-1)^(89781)

Now,

91743 and 89781 are both odd numbers as they end with 3 and 1, which are as we know odd numbers.

Thus,

(-1)^(91743) = (-1)

(-1)^(89781) = (-1)

Hence,

(-1)^(91743) + (-1)^(89781) = (-1) + (-1)

= (-2)

Therefore,

i^(183486) + i^(179562) = (-2)

(If you haven't understood the above solution I have as given written images of the solution, please do refer it then.)

Hope it helped and believing you understood it........All the best