The values of (16)1 / 4 are

Question

The values of (16)1 / 4 are (A) ± 2, ± 2 i (B) ± 4, ± 4 i (C) ± 1, ± i (D) None of these

Tardigrade Admin · Accepted Answer

We have (16)1 / 4 =(24)1 / 4=2(1)1 / 4 =2( cos 0+i sin 0)1 / 4 =2 cos (1/4)(2 k π+0)+i sin (1/4)(2 k π+0) k = 0, 1, 2, 3 =2 × 1,2 × i, 2 ×-1,2 ×-i=± 2, ± 2 i

Q. The values of $(16)^{1/4}$ are

$\pm 2, \pm 2 i$

$\pm 4, \pm 4 i$

$\pm 1, \pm i$

None of these

We have
$(16)^{1/4} = (2^{4})^{1/4} = 2 (1)^{1/4}$
$= 2 (cos 0 + i sin 0)^{1/4}$
$= 2 {cos \frac{1}{4} (2 kπ + 0) + i sin \frac{1}{4} (2 kπ + 0)},$
$k = 0, 1, 2, 3$
$= 2 \times 1, 2 \times i, 2 \times - 1, 2 \times - i = \pm 2, \pm 2 i$

Q. The values of (16)1/4 are

±2,±2i

±4,±4i

±1,±i