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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Explanation:

By definition, $i = \sqrt{-1}$

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

$i^{76} = i^0 = 1$