One of the values of -32i^25 is

Correct answer: (B) 4cis(3π)/5. Given, -32i^25 =32^25-i^25 =4-i^25 =40+isin3π2^25 =4cos3π2+isin3π2^25 Now using cosθ+isinθ^n=cosnθ+isinnθ we get, =4cos(3π)/2×2/5+isin(3π)/2×2/5 =4cos(3π)/5+isin(3π)/5 =4cis(3π)/5

Q49 / 160 · TS EAMCET 2022 (18 Jul Shift 2) · Mathematics

One of the values of ${(- 32 i)}^{\frac{2}{5}}$ is

A
$4 cis \frac{2 π}{5}$
Correct answer
B
$4 cis \frac{3 π}{5}$
C
$4 cis \frac{4 π}{5}$
D
$4 cis \frac{6 π}{5}$

Solution

Given,
${(- 32 i)}^{\frac{2}{5}}$
$= 32^{\frac{2}{5}} {(- i)}^{\frac{2}{5}}$
$= 4 {(- i)}^{\frac{2}{5}}$
$= 4 {(0 + i \sin \frac{3 π}{2})}^{\frac{2}{5}}$
$= 4 {(\cos \frac{3 π}{2} + i \sin \frac{3 π}{2})}^{\frac{2}{5}}$
Now using ${(\cos θ + i \sin θ)}^{n} = (\cos n θ + i \sin n θ)$ we get,
$= 4 (\cos \frac{3 π}{2} \times \frac{2}{5} + i \sin \frac{3 π}{2} \times \frac{2}{5})$
$= 4 (\cos \frac{3 π}{5} + i \sin \frac{3 π}{5})$
$= 4 cis \frac{3 π}{5}$

Hence, the correct answer is option: 'B'.

One of the values of -32i25 is

Solution

Relevant Questions

One of the values of ${(- 32 i)}^{\frac{2}{5}}$ is