Question

What does i^739 equal? explain how you found your answer.

Answer 1

Answer:   -i

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Explanation:

i = sqrt(-1)

Lets list out the first few powers of i

• i^0 = 1
• i^1 = i
• i^2 = -1
• i^3 = i*i^2 = i*(-1) = -i
• i^4 = (i^2)^2 = (-1)^2 = 1

By the time we reach the fourth power, we repeat the cycle over again (since i^0 is also equal to 1). The cycle is of length 4, which means we'll divide the exponent over 4 to find the remainder. Ignore the quotient. That remainder will determine if we go for i^0, i^1, i^2 or i^3.

For example, i^5 = i^1 because 5/4 leads to a remainder 1.

Another example: i^6 = i^2 since 6/4 = 1 remainder 2

Again, we only care about the remainder to find out which bin we land on.

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Turning to the question your teacher gave you, we have,

739/4 = 184 remainder 3

So i^739 = i^3 = -i

-i is the final answer

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Side notes:

• if i^a = i^b, then a-b is a multiple of 4
• Recall that the divisibility by 4 trick involves looking at the last two digits of the number. So i^739 is identical to i^39.