Final answer:
To simplify the expression with imag i³⁵, we note i has a cycle of powers repeating every four, so i³⁵ simplifies to i³ which is -i.
Step-by-step explanation:
To simplify the expression with imag i³⁵, we can use the cyclic property of powers of the imaginary unit i. Recall that i is defined as the square root of -1, and it has the following powers cycle:
- i = i
- i² = -1
- i³ = -i
- i⁴ = 1
Every fourth power of i is going to be equal to 1, and the powers of i repeat every 4. Therefore, i³⁵ can be simplified by dividing 35 by 4 and finding the remainder. 35 divided by 4 is 8 with a remainder of 3, so i³⁵ = i$⁴ imes i³ = i³ = -i. Therefore, the simplified form of imag i³⁵ is -i.