Solving Complex Number With Negative Fractional Exponent: i^(-21/2)

[Asawira Emaan]

Kindly help me with this.
Solve
i^(-21/2)
Note: i means iota.

 

[MarkFL]

I think I would begin by writing:

\(\displaystyle z=i^{-\frac{21}{2}}=\left(\cis\left(\frac{\pi}{2}\right)\right)^{-\frac{21}{2}}=\cis\left(-\frac{21\pi}{4}\right)=\cis\left(\frac{3\pi}{4}\right)=\frac{-1+i}{\sqrt{2}}\)

Does that make sense?

 

[I like Serena]

Kindly help me with this.
Solve
i^(-21/2)
Note: i means iota.

Hi Asawira Emaan, welcome to MHB!

Are you aware that $i=e^{\frac\pi 2 i}=\cos(\pi/2)+i\sin(\pi/2)$?
It means that we can calculate it as:
$$(i)^{-\frac{21}{2}} = (e^{\frac\pi 2 i})^{-\frac{21}{2}}
= e^{-\frac{21\pi}{4}i} = \cos\left(-\frac{21\pi}{4}\right)+i\sin\left(-\frac{21\pi}{4}\right)
$$

 

[Asawira Emaan]

I think I would begin by writing:

\(\displaystyle z=i^{-\frac{21}{2}}=\left(\cis\left(\frac{\pi}{2}\right)\right)^{-\frac{21}{2}}=\cis\left(-\frac{21\pi}{4}\right)=\cis\left(\frac{3\pi}{4}\right)=\frac{-1+i}{\sqrt{2}}\)

Does that make sense?

Hi Asawira Emaan, welcome to MHB!

Are you aware that $i=e^{\frac\pi 2 i}=\cos(\pi/2)+i\sin(\pi/2)$?
It means that we can calculate it as:
$$(i)^{-\frac{21}{2}} = (e^{\frac\pi 2 i})^{-\frac{21}{2}}
= e^{-\frac{21\pi}{4}i} = \cos\left(-\frac{21\pi}{4}\right)+i\sin\left(-\frac{21\pi}{4}\right)
$$

but the answer of this question is -i (negative iota)

 

[I like Serena]

but the answer of this question is -i (negative iota)

It looks as if you were given the wrong answer then.
It's as MarkFL wrote, the answer is $\frac{-1+i}{\sqrt 2}$.

 

[MarkFL]

but the answer of this question is -i (negative iota)

It is true that:

\(\displaystyle i^{-21}=-i\)

But:

\(\displaystyle i^{-\frac{21}{2}}\ne-i\)

 

[Asawira Emaan]

Oh yes!
Thanks for help