Question

i^9 + i^19 in the form of a+ib​

Answer 1

i⁹ + i¹⁹ = -4+i +( -8+3i)

Explanation:

To find:

i⁹ + i¹⁹ in the form of a+ib​

We know that,

i² = -1

==> Split i⁹ in terms of i²

Add the power values

==> 9 = 2+2+2+2+1

==> 9 =8+1

==> 9= 9

==> i⁹ = i²+i²+i²+i²+i

==> i⁹ = -1+(-1)+(-1)+(-1)+i

==> i⁹ = -1-1-1-1+i

==> i⁹ = -4+i

Split  i¹⁹

==> We know i⁹ so, split into i⁹

==> i¹⁹ = i⁹+ i⁹+i

Add the power values

==> 19 = 9+9+1

==> 19 = 18+1

==> 19 =19

==> i¹⁹ = -4+i +(-4+i)+i

==> i¹⁹ = -4+i-4+i+i

==> i¹⁹ = -8+3i

i⁹ + i¹⁹ = -4+i +( -8+3i)

i⁹ + i¹⁹ = -4 + i -8+3i

i⁹ + i¹⁹ = -4-8+i+3i

i⁹ + i¹⁹ = -12+4i

a+ib = -12+i4