Solve and simplify
(3+9i)-(12-5i)
8+6i/7-5i
Rewrite as:
3+9i-12+5i
Add like terms:
-9+14i is your answer.
for the 2nd, multiply top and bottom by 7+5i
(8+6i)/(7-5i) * (7+5i)/(7+5i)
=(56 + 40i + 42i + 30i^2)/(48 - 25i^2)
= (26 + 82i)/73 . remember i^2 = -1
To solve and simplify (3+9i)-(12-5i), follow these steps:
1. Distribute the negative sign to each term inside the parentheses:
(3 + 9i) - 12 + 5i
2. Combine like terms:
3 - 12 + 9i + 5i
3. Combine the real (constant) terms separately from the imaginary terms:
(3 - 12) + (9i + 5i)
-9 + 14i
Therefore, the simplified form of (3+9i)-(12-5i) is -9 + 14i.
Now, let's solve and simplify (8+6i)/(7-5i):
1. To divide two complex numbers, we multiply both the numerator and denominator by the conjugate of the denominator. The conjugate of a complex number a + bi is a - bi.
2. The conjugate of (7-5i) is (7+5i), so we multiply both the numerator and denominator by (7+5i):
(8 + 6i)(7 + 5i) / (7 - 5i)(7 + 5i)
3. Multiply the numerators and denominators using the FOIL method (First, Outer, Inner, Last):
(8 * 7 + 8 * 5i + 6i * 7 + 6i * 5i) / (7 * 7 + 7 * 5i - 5i * 7 - 5i * 5i)
4. Simplify each term:
(56 + 40i + 42i + 30i^2) / (49 + 35i - 35i - 25i^2)
5. Simplify the terms with the imaginary unit i (i^2 = -1):
(56 + 82i + 30 * -1) / (49 - 25 * -1)
6. Simplify further:
(56 + 82i - 30) / (49 + 25)
7. Combine like terms:
(26 + 82i) / 74
8. Divide each term by 74:
26/74 + 82i/74
9. Simplify the fraction:
13/37 + (41/37)i
Therefore, the simplified form of (8+6i)/(7-5i) is 13/37 + (41/37)i.