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 please explain
similar to "rationalizing the denominator"
Multiply top and bottom by 2+i and remember that i^2 = -1
(1+i)/(2-i) x (2+i)/(2+i)
= (2+i+2i+i^2)/(4-i^2)
= (2+3i-1)/(4-(-1))
= (1+3i)/5
btw, the choice that fits my answer would be D, they simply split it into 2 parts.
I don't know why they would do that
To simplify the expression (1+i)/(2-i), we need to rationalize the denominator.
To do this, we can multiply both the numerator and denominator by the conjugate of the denominator.
The conjugate of a complex number a+bi is a-bi.
So, in this case, the conjugate of 2-i is 2+i.
Now, let's multiply the numerator and denominator by 2+i:
(1+i)/(2-i) * (2+i)/(2+i)
Expanding this, we get:
((1+i)*(2+i))/((2-i)*(2+i))
Simplifying the numerators and denominators separately:
Numerator: (1+i)*(2+i) = 1(2)+1(i)+2(i)+i(i) = 2+i+2i-i^2
Now, i^2 = -1, so we can substitute it:
2+i+2i-(-1) = 2+i+2i+1 = 3+3i
Denominator: (2-i)*(2+i) = 2(2)-2(i)+2(i)-i(i) = 4+2i-2i-i^2
Again, substituting i^2 = -1:
4+2i-2i-(-1) = 4+2i-2i+1 = 5
Now we have:
(3+3i)/5
This expression is equal to (1/5)(3+3i).
So, the simplified form of (1+i)/(2-i) is (1/5)(3+3i).
Therefore, the answer is D) (1/5)+(3/5i).