Let x=2013, y=140542, z=−142555. Find
[(x−y)/z + (y−z)/x + (z−x)/y]*[x/(y−z) + y/(z−x) + z/(x−y)].
To find the value of the expression [(x−y)/z + (y−z)/x + (z−x)/y]*[x/(y−z) + y/(z−x) + z/(x−y)], we will first simplify each term separately and then perform the arithmetic operations.
Let's start with the first term, (x−y)/z. We substitute the given values of x, y, and z into the expression:
(x−y)/z = (2013−140542)/−142555
= −138529/−142555
To simplify this fraction, we can divide both the numerator and denominator by their greatest common divisor (GCD), which in this case is 1:
−138529/−142555
= 138529/142555
Similarly, for the second term, (y−z)/x:
(y−z)/x = (140542−(−142555))/2013
= 283097/2013
And for the third term, (z−x)/y:
(z−x)/y = ((−142555)−2013)/140542
= −144568/140542
Now, let's move on to the second part of the expression: x/(y−z), y/(z−x), and z/(x−y).
x/(y−z) = 2013/(140542−(−142555))
= 2013/283097
y/(z−x) = 140542/((−142555)−2013)
= 140542/144568
z/(x−y) = (−142555)/(2013−140542)
= −142555/−138529
= 142555/138529
Now, we have all the terms simplified. Let's substitute these values back into the original expression:
[(138529/142555) + (283097/2013) + (−144568/140542)] * [(2013/283097) + (140542/144568) + (142555/138529)]
Before performing the arithmetic operations, we can simplify the fractions further if possible. However, since the numerator and denominator of each fraction are relatively prime, we cannot reduce them any further.
Substituting these values into the expression, we get:
[(138529/142555) + (283097/2013) + (−144568/140542)] * [(2013/283097) + (140542/144568) + (142555/138529)]
= (138529/142555 + 283097/2013 + −144568/140542) * (2013/283097 + 140542/144568 + 142555/138529)
Now, we can multiply the numerator of the first term with the numerator of the second term and the denominator of the first term with the denominator of the second term, separately.
= (138529 * 2013 + 283097 * 140542 + −144568 * 142555) / (142555 * 283097 * 140542) * (2013 * 138529 + 140542 * 144568 + 142555 * 283097) / (283097 * 144568 * 138529)
Evaluating the arithmetic operations will yield the final result.