Express \(\frac{2+3i}{3-3i}\) in the form \(a+bi\).
Answer (2)
( 2 + 3 i ) : ( 3 − 3 i ) = 3 − 3 i 2 + 3 i ⋅ 3 + 3 i 3 + 3 i = ( 3 − 3 i ) ( 3 + 3 i ) ( 2 + 3 i ) ( 3 + 3 i ) = 3 2 − ( 3 i ) 2 6 + 6 i + 9 i − 9 = 9 + 9 − 3 + 15 i = 18 − 3 + 15 i = − 18 3 + 18 15 i = − 6 1 + 6 5 i
The expression 3 − 3 i 2 + 3 i can be expressed in the form a + bi as − 6 1 + 6 5 i after multiplying by the conjugate of the denominator. This eliminates the imaginary number in the denominator and allows us to separate the real and imaginary parts. The final result shows that the real part is − 6 1 and the imaginary part is 6 5 .
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