To simplify this expression, we first need to rationalize the denominators of the fractions.
Rationalize the denominator of the first fraction: 5 - √7
Multiply both the numerator and denominator of the first fraction by the conjugate of the denominator: 9/(5-√7) * (5+√7)/(5+√7) = 9(5+√7)/(25 - 7) = 9(5+√7)/18 = (5+√7)/2
Rationalize the denominator of the second fraction: 7 + √5
Multiply both the numerator and denominator of the second fraction by the conjugate of the denominator: 22/(7+√5) * (7-√5)/(7-√5) = 22(7-√5)/(49 - 5) = 22(7-√5)/44 = (7-√5)/2
Rationalize the denominator of the third fraction: √7 + √5
Multiply both the numerator and denominator of the third fraction by the conjugate of the denominator: 1/(√7+√5) * (√7-√5)/(√7-√5) = (√7-√5)/(7 - 5) = (√7-√5)/2
Now our expression simplifies to: (5+√7)/2 + (7-√5)/2 - (√7-√5)/2
Combine the terms with the same denominator: [(5+√7) + (7-√5) - (√7-√5)] / 2 (5 + √7 + 7 - √5 - √7 + √5) / 2 (12) / 2 6
To simplify this expression, we first need to rationalize the denominators of the fractions.
Rationalize the denominator of the first fraction: 5 - √7Multiply both the numerator and denominator of the first fraction by the conjugate of the denominator:
Rationalize the denominator of the second fraction: 7 + √59/(5-√7) * (5+√7)/(5+√7) = 9(5+√7)/(25 - 7) = 9(5+√7)/18 = (5+√7)/2
Multiply both the numerator and denominator of the second fraction by the conjugate of the denominator:
Rationalize the denominator of the third fraction: √7 + √522/(7+√5) * (7-√5)/(7-√5) = 22(7-√5)/(49 - 5) = 22(7-√5)/44 = (7-√5)/2
Multiply both the numerator and denominator of the third fraction by the conjugate of the denominator:
1/(√7+√5) * (√7-√5)/(√7-√5) = (√7-√5)/(7 - 5) = (√7-√5)/2
Now our expression simplifies to:
(5+√7)/2 + (7-√5)/2 - (√7-√5)/2
Combine the terms with the same denominator:
[(5+√7) + (7-√5) - (√7-√5)] / 2
(5 + √7 + 7 - √5 - √7 + √5) / 2
(12) / 2
6
Therefore, the simplified expression is 6.