Question 496646
 i<sup>14</sup>
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All you have to know in all problem like this are these 5 facts: 

1.  i<sup>2</sup> = -1
2.  (-1)<sup>even</sup> = 1
3.  (-1)<sup>odd</sup> = -1
4.  A<sup>B×C</sup> = (A<sup>B</sup>)<sup>C</sup> 
5.  A<sup>B+C</sup> = A<sup>B</sup>A<sup>C</sup>

i<sup>14</sup> =  i<sup>2×7</sup> = (i<sup>2</sup>)<sup>7</sup> = (-1)<sup>7</sup> = -1

Some other examples:

i<sup>682</sup> = i<sup>2×341</sup> = (i<sup>2</sup>)<sup>341</sup> = (-1)<sup>341</sup> = -1

i<sup>796</sup> = i<sup>2×398</sup> = (i<sup>2</sup>)<sup>398</sup> = (-1)<sup>398</sup> = 1

For odd powers of i, first write the exponent as an even number + 1:

i<sup>47</sup> = i<sup>46+1</sup> = i<sup>46</sup>i<sup>1</sup> = i<sup>2×23</sup>i = (i<sup>2</sup>)<sup>23</sup>i = (-1)<sup>23</sup>i = -1i = -i

i<sup>109</sup> = i<sup>108+1</sup> = i<sup>108</sup>i<sup>1</sup> = i<sup>2×54</sup>i = (i<sup>2</sup>)<sup>54</sup>i = (-1)<sup>54</sup>i = 1i = i

Edwin</pre>