Answer: Sure, I can help you with that. The expression you provided is a division of two binomials, which is in the form of (a + b) / (c - d). To rationalize this expression, we need to multiply both the numerator and denominator by the conjugate of the denominator, which is (7 + 3\sqrt{5}) * (7 + 3\sqrt{5}) = 49 + 42\sqrt{5} + 45 = 94 + 42\sqrt{5}. This gives us:
(7 + 3\sqrt{5}) * (7 + 3\sqrt{5}) = 49 + 42\sqrt{5} + 45 = 94 + 42\sqrt{5}
(7 - 3\sqrt{5}) * (7 + 3\sqrt{5}) = 49 - 45 = 4
Therefore, the rationalized form of the expression is:
(7 + 3\sqrt{5}) * (7 + 3\sqrt{5}) / ((7 - 3\sqrt{5}) * (7 + 3\sqrt{5})) = (94 + 42\sqrt{5}) / 4
Simplifying this expression gives us:
(94 + 42\sqrt{5}) / 4 = (47 + 21\sqrt{5}) / 2
So, the rationalized form of 7+3\sqrt{5} by 7-3\sqrt{5} is (47 + 21\sqrt{5}) / 2.