express the complex number in standard form(that is like a+bi)

1/2+i

2+i/2-i

rationalize the denominator. I will do one:

(2+i)/(2-i)= 2+i)(2+i)/(2-i)(2+i)=

(2+i)^2/(2-i^2)= (4+4i+i^2)/3
you can complete it.

so is that the final answer

os is the final answer 8i+i^3/3

(2+i)/(2-i)

=(2+i)(2+i)/((2+i)(2-i))
=(4+4i+i²)/(2²-i²)
=(4+4i-1)/(4-(-1))
=(3+4i)/5

You can proceed along the same lines for the first problem.

To express a complex number in standard form, which is in the form of "a+bi," where a and b are real numbers, you need to simplify the given expressions.

1. Expressing 1/2 + i in standard form:
To simplify this expression, we can multiply the numerator and denominator by the conjugate of the denominator to eliminate the complex denominator.

The conjugate of (2+i) is (2-i). So, we multiply the numerator and denominator of 1/2 + i by (2-i):

(1/2 + i) * (2-i) / (2+i) * (2-i)

Expanding the numerator and denominator, we get:
((1/2)*(2) + (1/2)*(-i) + (i)*(2) + (i)*(-i)) / ((2)*(2) + (2)*(-i) + (2)*(i) + (i)*(-i))

Simplifying further:
(1 + (-1/2)i + 2i - i^2) / (4 + (-2i) + (2i) - i^2)

Substituting i^2 as -1:
(1 + (-1/2)i + 2i - (-1)) / (4 + (-2i) + (2i) - (-1))

Rearranging and combining like terms:
(1 + 7/2i) / (4 + (-2i) + (2i) + 1)
(1 + 7/2i) / (5 + 0i)

Now, to divide by a complex number, we can multiply both the numerator and denominator by the conjugate of the denominator to eliminate the complex denominator:

(1 + 7/2i) * (5 - 0i) / (5 + 0i) * (5 - 0i)
(5 + 7/2i - 0i - 0i) / (5^2 + 0i - 0i - 0)
(5 + 7/2i) / 25

The final answer is:
(5 + 7/2i) / 25, which can be written as:
5/25 + (7/2i) / 25
1/5 + (7/50)i

Therefore, the complex number 1/2 + i in standard form is written as 1/5 + (7/50)i.

2. Expressing (2+i)/(2-i) in standard form:
To simplify this expression, we can multiply the numerator and denominator by the conjugate of the denominator, which is (2+i). By doing so, we can eliminate the complex denominator.

(2+i)/(2-i) * (2+i)/(2+i)
((2+i)*(2+i)) / ((2-i)*(2+i))

Expanding the numerator and denominator, we get:
(4 + 4i + i^2) / (4 - i^2)

Since i^2 is -1, we substitute it:
(4 + 4i + (-1)) / (4 - (-1))

Rearranging and combining like terms:
(3 + 4i) / (4 + 1)

Dividing both the numerator and denominator by 1:
(3 + 4i) / 5

The final answer is:
(3 + 4i)/5, which can be written as:
3/5 + (4/5)i

Therefore, the complex number (2+i)/(2-i) in standard form is 3/5 + (4/5)i.