Express (5−3i)3 in the form a+i b.
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Step-by-Step Solution
Step 1: Expand the expression
We need to expand the given expression (5−3i)3. This can be done using the binomial expansion formula (a−b)3=a3−3a2b+3ab2−b3. Here, a=5 and b=3i.
Step 2: Calculate powers of terms
Now, we calculate the powers of the individual terms. Remember that i2=−1 and i3=i2⋅i=−1⋅i=−i.
Step 3: Substitute and simplify
Substitute the calculated powers back into the expanded expression. Then, perform the multiplications and simplify the terms.
Step 4: Combine real and imaginary parts
Finally, group the real parts and the imaginary parts together. Combine them to express the result in the form a+ib.