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

#### Asawira Emaan

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

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#### MarkFL

Staff member
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?

#### Klaas van Aarsen

##### MHB Seeker
Staff member
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

##### New member
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)

#### Klaas van Aarsen

##### MHB Seeker
Staff member
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

Staff member
but the answer of this question is -i (negative iota)
It is true that:

$$\displaystyle i^{-21}=-i$$

But:

$$\displaystyle i^{-\frac{21}{2}}\ne-i$$

Oh yes!
Thanks for help