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.

Express $(5-3 i)^{3}$ in the form $a+i b$ .

Answer: $-10 - 198i$

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 $(a-b)^3 = a^3 - 3a^2b + 3ab^2 - b^3$ . Here, $a=5$ and $b=3i$ .

Step 2: Calculate powers of terms

Now, we calculate the powers of the individual terms. Remember that $i^2 = -1$ and $i^3 = i^2 \cdot i = -1 \cdot i = -i$ .

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 $a+ib$ .

Express (5−3i)3(5-3 i)^{3}(5−3i)3 in the form a+iba+i ba+ib.