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Question
Write (i25)3 in polar form.
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Solution
\[\left( i^{25} \right)^3 = i^{75} \]
\[ = i^{4 \times 18 + 3} \]
\[ = \left( i^4 \right)^{18} . i^3 \]
\[ = i^3 [ \because i^4 = 1]\]
\[ = - i [ \because i^3 = - i]\]
Let \[z = 0 - i\]
Then,
\[\left| z \right| = \sqrt{0^2 + \left( - 1 \right)^2} = 1\].
Let θ be the argument of z and α be the acute angle given by
\[\tan\alpha = \frac{1}{0} = \infty \]
\[ \Rightarrow \alpha = \frac{\pi}{2}\]
Clearly, z lies in fourth quadrant. So, arg(z) =
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