Simplify the following quotient of complex numbers into the form a+bi
1+8i
----
-6+i
(1+8i)(-6-i)/[ (-6+i)(-6-i)]
(-6 - 49 i + 8) / (36 +1)
(2 -49 i)/37
2/37 - (49/37) i
To simplify the quotient of complex numbers, we can use a technique called rationalizing the denominator. Here's how you can simplify the given quotient:
Step 1: Rationalize the denominator
Multiply both the numerator and denominator by the conjugate of the denominator. The conjugate of -6+i is -6-i.
(1+8i) * (-6-i) / (-6+i) * (-6-i)
Step 2: Simplify the numerator and denominator
To simplify the numerator, we can distribute the terms and combine like terms:
(1+8i) * (-6-i) = -6 - i - 48i - 8i^2
= -6 - 49i - 8(-1) (since i^2 = -1)
= -6 - 49i + 8
= 2 - 49i
To simplify the denominator, we multiply the conjugates:
(-6+i) * (-6-i) = 36 - i^2
= 36 - (-1)
= 36 + 1
= 37
Step 3: Rewrite the quotient
Now write the simplified form of the quotient by dividing the numerator by the denominator:
(2 - 49i) / 37
So the simplified form of the given quotient is (2 - 49i) / 37.