Question

Match each complex number with its equivalent expression i^157 i^315 i^102 i^76

Answer 1

Answers:

`i^(157) = i\\\\i^(315) = -i\\\\i^(102) = -1\\\\i^(76) = 1\\\\`

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Step-by-step explanation:

By definition,
`i = √(-1)`

Squaring both sides gets us
`i^2 = -1`

Then multiply both sides by i to get
`i^3 = -i`

Repeat the last step and you should get
`i^4 = -i^2 = -(-1) = 1`

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Notice we have this pattern going on:

`i^0 = 1\\\\i^1 = i\\\\i^2 = -1\\\\i^3 = -i\\\\i^4 = 1\\\\`

Once we reach i^4, we start the process over again.

It repeats every 4 terms.

This means we'll divide the exponent over 4 and look at the remainder. We ignore the quotient completely.

157/4 = 39 remainder 1

That remainder 1 is the exponent of the simplified term

`i^(157) = i^1 = i`

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Similarly,

315/4 = 78 remainder 3

So
`i^(315) = i^3 = -i`

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102/4 = 25 remainder 2

`i^(102) = i^2 = -1`

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76/4 = 19 remainder 0

`i^(76) = i^0 = 1`