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Question

If z=7-i3-4i, then z14 is equal to


A

27

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B

27i

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C

214i

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D

-27i

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E

-214

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Solution

The correct option is D

-27i


Explanation for the correct option.

Step 1. Simplify the complex number z.

In z=7-i3-4i, multiply the numerator and denominator by 3+4i.

z=7-i3-4i·3+4i3+4i=7-i3+4i3-4i3+4i=21+28i-3i-4i29-16i2=21+25i-4(-1)9-16(-1)i2=-1=21+25i+49+16=25+25i25=1+i

Step 2. Find the value of Z2

So, Z=1+i. Now Z2 is given as:

Z2=(1+i)2=1+2i+i2=1+2i+(-1)i2=-1=1+2i-1=2i

Step 3. Find the value of Z14.

Now, Z14 is given as:

Z14=Z27=2i7=27i7=27i4+3=27i4i3am+n=aman=27×1×-ii4=1,i3=-i=-27i

So the value of z14 is -27i.

Hence, the correct option is D.


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