Simplify (4-2i/-10+5i) x (3-i/-3+i)
To simplify the expression (4-2i/-10+5i) x (3-i/-3+i), we can use the concept of complex conjugates.
Step 1: Find the complex conjugates of the denominators.
The complex conjugate of -10+5i is -10-5i.
The complex conjugate of -3+i is -3-i.
Step 2: Multiply both the numerator and denominator of the first fraction by the conjugate of the second denominator, and vice versa.
So, we have:
[(4-2i)(-3-i)] / [(-10+5i)(-3-i)]
Step 3: Simplify the numerator and denominator.
For the numerator:
(4-2i)(-3-i) = -12 + 6i + 6i + 2i^2
Simplifying further:
-12 + 12i - 2
-14 + 12i
For the denominator:
(-10+5i)(-3-i) = 30 +3i -15i -5i^2
Simplifying further:
30 - 12i - 5(-1)
30 - 12i + 5
35 - 12i
Step 4: Simplify the result.
[(4-2i)(-3-i)] / [(-10+5i)(-3-i)] = (-14 + 12i) / (35 - 12i)
To rationalize the denominator, we multiply the numerator and denominator by the conjugate of the denominator:
((-14 + 12i)(35 + 12i)) / ((35 - 12i)(35 + 12i))
Expanding further:
(-14*35) + (-14*12i) + (12i*35) + (12i*12i) / (35*35) + (35*12i) + (-12i*35) + (-12i*12i)
Simplifying:
-490 - 168i + 420i +144 / 1225 + 420i - 420i - 144i^2
Simplifying further:
-490 + 252i + 144 / 1225 - 144i^2
Since i^2 is equal to -1, we can substitute:
-490 + 252i + 144 / 1225 - 144(-1)
Simplifying:
-490 + 252i + 144 / 1225 + 144
Combining like terms:
(-346 + 252i) / 1369
So, the simplified expression is (-346 + 252i) / 1369.