Question:
If $z=\frac{7-i}{3-4i} \, then\, z^{14}=$
If $z=\frac{7-i}{3-4i} \, then\, z^{14}=$
Updated On: Aug 15, 2024
- $2^{7}$
- $2^{7} i$
- $2^{14} i$
- $-2^{7}i$
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The Correct Option is D
Solution and Explanation
$z=\frac{7- i}{3-4 i} \times\frac{3+4i}{3+4i}$
$=\frac{21+25 i+4}{16+9}=\frac{25\left(1+i\right)}{25}=\left(1+i\right)$
$z^{14}=\left(1+i\right)^{14}=\left[\left(1+i\right)^{2}\right]^{7}=\left(2i\right)^{7}=2^{7} i^{7}=-2^{7} i$
$=\frac{21+25 i+4}{16+9}=\frac{25\left(1+i\right)}{25}=\left(1+i\right)$
$z^{14}=\left(1+i\right)^{14}=\left[\left(1+i\right)^{2}\right]^{7}=\left(2i\right)^{7}=2^{7} i^{7}=-2^{7} i$
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