To simplify the expression 2 / (2 + 5i), we can multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of a complex number a + bi is a – bi. In this case, the conjugate of 2 + 5i is 2 – 5i.
So, let’s rewrite the expression:
2 / (2 + 5i) × (2 – 5i) / (2 – 5i)
This gives us:
(2 × (2 – 5i)) / ((2 + 5i) × (2 – 5i))
Now, calculate the numerator:
2 × (2 – 5i) = 4 – 10i
Next, for the denominator, we use the difference of squares formula:
(2 + 5i) × (2 – 5i) = 2² – (5i)² = 4 – 25(-1) = 4 + 25 = 29
Putting it all together, we have:
(4 – 10i) / 29
This can be separated into real and imaginary parts:
(4/29) – (10/29)i
Thus, the simplified form of the expression 2 / (2 + 5i) is:
4/29 – (10/29)i