Divide and simplify to the form a+bi.
(-8+i)/(2-3i)
Is the answer (23/13)+(43/13)i
(-8+i)/(2-3i) * (2+3i)/(2+3i)
= (-16 -24i +2i +3i^2)/(4-9i^2)
= (-19 -22i)/13 or -19/13 - 22i/13
To divide and simplify complex numbers of the form (a+bi)/(c+di) to the form a+bi, you can use the technique called "rationalizing the denominator."
Step 1: Multiply the numerator and denominator by the conjugate of the denominator. The conjugate of a complex number with the form a+bi is a-bi.
In this case, the denominator is 2-3i, so the conjugate is 2+3i. So we can rewrite the expression as:
((-8+i)/(2-3i)) * ((2+3i)/(2+3i))
Step 2: Simplify the expression.
Let's first simplify the numerator:
((-8+i)*(2+3i)) = (-8*2) + (-8*3i) + (i*2) + (i*3i)
= -16 - 24i + 2i + 3i^2
= -16 - 22i + 3i^2
Now, let's simplify the denominator:
(2-3i)*(2+3i) = (2*2) + (2*3i) + (-3i*2) + (-3i*3i)
= 4 + 6i - 6i - 9i^2
= 4 + 6i - 6i - 9(-1)
= 4 + 6i - 6i + 9
= 13
So now, we have:
((-16 - 22i + 3i^2)/13)
Step 3: Further simplify the expression.
Since i^2 is equal to -1, we can substitute it in:
((-16 - 22i - 3)/13)
= (-19 - 22i)/13
Therefore, the simplified form is (-19/13) + (-22/13)i.
So, the given answer, (23/13) + (43/13)i, is incorrect. The correct answer is (-19/13) + (-22/13)i.