If (8+√19)/(8−√19)=(a/c)+[(b/c)*√19], where a and c are coprime positive integers, and b and c are coprime positive integers, what is a+b+c?
To solve this problem, we can use the concept of rationalizing the denominator to simplify the expression (8+√19)/(8−√19).
Step 1: Rationalize the denominator
To rationalize the denominator, we multiply both the numerator and denominator by the conjugate of the denominator. In this case, the conjugate of (8−√19) is (8+√19):
[(8+√19) * (8+√19)] / [(8−√19) * (8+√19)]
Simplifying the denominator, (8−√19) * (8+√19) can be expanded using the difference of squares formula, which states that (a−b) * (a+b) = a^2−b^2:
[(8+√19) * (8+√19)] / (8^2 − (√19)^2)
= (8+√19)*(8+√19) / (64 - 19)
= (64 + 16√19 + 19) / 45
= (83 + 16√19) / 45
Step 2: Match the given form
Now we need to match the expression obtained [(83 + 16√19) / 45] with the given form (a/c) + [(b/c)*√19].
Comparing the expressions, we can see that:
a = 83, b = 16, and c = 45.
Step 3: Find a+b+c
To find a+b+c, we simply add the values of a, b, and c:
a + b + c = 83 + 16 + 45
= 144
Therefore, a+b+c = 144.