Simplify: (1+i)/(2-i)

A)(1/3)+(2/3i)
B)(1/5)+(2/5i)
C)(1/3)+(i)
D)(1/5)+(3/5i)
I have no clue

To simplify the fraction (1+i)/(2-i), we can use the concept of complex conjugates.

Step 1: Find the complex conjugate of the denominator.
The complex conjugate of 2-i is 2+i.

Step 2: Multiply both the numerator and denominator by the complex conjugate of the denominator.
(1+i)(2+i) / (2-i)(2+i)

Step 3: Simplify the numerator and denominator individually.

Numerator:
(1+i)(2+i) = (1*2) + (1*i) + (1*2*i) + (1*i*i)
= 2 + i + 2i + i^2
= 2 + 3i - 1
= 1 + 3i

Denominator:
(2-i)(2+i) = (2*2) + (2*i) - (i*2) - (i*i)
= 4 + 2i - 2i - i^2
= 4 - 1
= 3

Step 4: Simplify the fraction.
(1 + 3i) / 3

Therefore, the simplified form of (1+i)/(2-i) is (1 + 3i) / 3.

Now, let's compare the simplified form to the options provided:

A) (1/3)+(2/3i)
B) (1/5)+(2/5i)
C) (1/3)+(i)
D) (1/5)+(3/5i)

The simplified form (1 + 3i) / 3 does not match any of the given options. It is possible that none of the options provided are correct or there may be a mistake in the options.

Please check the options again or consider seeking further clarification.