Problem:
Simplify the expression 5i/(1-i)(2-i)(3-i).

Solution:

To simplify the given expression, we can start by simplifying each denominator individually and then multiplying them together. Let's begin step by step.

Step 1: Simplifying the first denominator (1-i):
To simplify 1-i, we can multiply both the numerator and denominator by the conjugate of the denominator, which is 1+i. Multiplying the conjugate eliminates the imaginary part in the denominator.

1-i * (1+i)/(1+i) = (1-i+i-i^2)/(1^2-i^2)
= (1+1+1)/(1+1)
= 3/2

So, the first denominator simplifies to 3/2.

Step 2: Simplifying the second denominator (2-i):
Similar to the first step, we multiply both the numerator and denominator by the conjugate of the denominator, which is 2+i, to eliminate the imaginary part.

2-i * (2+i)/(2+i) = (4+2i-2i-i^2)/(2^2-i^2)
= (4+2i-2i-1)/(4+1)
= 3/5

Therefore, the second denominator simplifies to 3/5.

Step 3: Simplifying the third denominator (3-i):
Again, we multiply both the numerator and denominator by the conjugate of the denominator, which is 3+i.

3-i * (3+i)/(3+i) = (9+3i-3i-i^2)/(3^2-i^2)
= (9+3i-3i-1)/(9+1)
= 8/10
= 4/5

So, the third denominator simplifies to 4/5.

Step 4: Simplifying the numerator:
The numerator is 5i, which doesn't require any further simplification.

Step 5: Simplifying the entire expression:
Now that we have simplified each denominator, we can multiply them together along with the numerator.

5i/(1-i)(2-i)(3-i) = 5i/(3/2)(3/5)(4/5)
= 5i * (2/3) * (5/3) * (5/4)
= (25/3) * (5/4) * i

Final Answer:
The expression 5i/(1-i)(2-i)(3-i) simplifies to (25/3) * (5/4) * i, or (125/12) * i.

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