If Z -13-5 i -4-9 i find z 6-

Rationalize the complex number by multiplying and dividing by 4+9i 13 5i/4 9i × 4+9i/4+9i = 52 20i+117i 45i^2/16 i^2 81 =97+97i/97= (1+i) In polar form z=√(2) ...

You visited us 1 times! Enjoying our articles? Unlock Full Access!

Byju's Answer

Standard XI

Mathematics

Purely Imaginary

If Z =13-5 i ...

Question

If Z= $\frac{(13 - 5 i)}{(4 - 9 i)}$ find $z^{6}$

__

Open in App

Solution

Rationalize the complex number by multiplying and dividing by 4+9i

$\frac{13 - 5 i}{4 - 9 i}$ $\times$ $\frac{4 + 9 i}{4 + 9 i}$ = $\frac{52 - 20 i + 117 i - 45 i^{2}}{16 - i^{2} 81}$

= $\frac{97 + 97 i}{97}$ = (1+i)

In polar form

z= $\sqrt{2}$ (cos $\frac{π}{4}$ +isin $\frac{π}{4}$ )

$z^{6}$ = [ $\sqrt{2}$ $(c o s \frac{π}{4} + i s i n \frac{π}{4})]^{6}$

$(\sqrt{2})^{6}$ [ $(c o s \frac{π}{4} + i s i n \frac{π}{4})]^{6}$

$z^{6}$ = 8[cos $\frac{3 π}{2}$ + isin $\frac{3 π}{2}$ ]

=8[0-i] = -8i

Suggest Corrections

Similar questions

If Z = $\frac{13 - 5 i}{4 - 9 i}$ can be written in a+ib. Find the value of a+b.

___

Q. If

{(2 + z)}^{6} + {(2 - z)}^{6} = 0

and

ω = \frac{2 + z}{2 - z}

If |z - 5i| = |z + 5i|, then find the locus of z.

Q. If

|z - 5 i| = |z + 5 i|

, then find the locus of z.

If 9 \equiv 3 (m o d 6) and 14 \equiv 2 (m o d 6)

, then

126 \equiv z (m o d 6)

. Find z.

Join BYJU'S Learning Program

If Z= (13−5i)(4−9i) find z6 __

Rationalize the complex number by multiplying and dividing by 4+9i 13−5i4−9i × 4+9i4+9i = 52−20i+117i−45i216−i281 =97+97i97= (1+i) In polar form z=√2 (cosπ4+isinπ4) z6 = [√2(cosπ4+isinπ4)]6 (√2)6[(cosπ4+isinπ4)]6 z6= 8[cos 3π2 + isin3π2] =8[0-i] = -8i

If Z= $\frac{(13 - 5 i)}{(4 - 9 i)}$ find $z^{6}$

__