i) just add the real and imaginary parts separately
ii) rationalize the denominator
(3+4i)/(6+3i) * (6-3i)/(6-3i)
= (3+4i)(6-3i)/(6^2+3^2)
...
i) (4+3i) + (6+4i)
ii) 3 + 4i/6+3i
ii) rationalize the denominator
(3+4i)/(6+3i) * (6-3i)/(6-3i)
= (3+4i)(6-3i)/(6^2+3^2)
...
i) (4+3i) + (6+4i):
To add complex numbers, you simply add their real parts (the numbers without "i") and their imaginary parts (the numbers with "i").
Real part: (4 + 6) = 10
Imaginary part: (3 + 4) = 7
So, the sum is 10 + 7i.
Expressing it in standard form means writing it in the form "a + bi" where a and b are real numbers, and "i" is the imaginary unit. Therefore, the standard form for the sum is 10 + 7i.
ii) 3 + 4i / 6 + 3i:
To divide complex numbers, we multiply both the numerator and denominator by the complex conjugate of the denominator. The complex conjugate is obtained by changing the sign of the imaginary part of the denominator.
Complex conjugate of 6 + 3i is 6 - 3i.
Now, let's multiply the numerator and denominator by the complex conjugate:
(3 + 4i) * (6 - 3i) / (6 + 3i) * (6 - 3i)
Simplifying this expression, we get:
(18 - 9i + 24i - 12i^2) / (36 - 9i^2)
Remember that i^2 is equal to -1. So, simplifying further:
(6 + 15i) / (36 + 9)
(6 + 15i) / 45
Dividing both the real and imaginary parts by 45:
6/45 + 15i/45
Simplifying this expression:
2/15 + 1/3i
So, the result in standard form is 2/15 + 1/3i.