7.
rationalize the denominator of 5/√3:
5/√3
= (5√3) / (√3√3)
= (a) (5√3) / 3
8.
rationalize the denominator of 4/(5+√2):
4 / (5 + √2)
= (4(5 - √2)) / ((5 + √2)(5 - √2)) [use it's conjugate (a + b)(a - b) = a² - b²]
= (20 - 4√2) / (25 - 2)
= (c) (20 - 4√2) / 23
9.
we choose the binomial:
5 + √3
it's conjugate is:
5 - √3
because the conjugate of (a + b) is (a - b)
we multiply the binomial by its conjugate:
(5 + √3)(5 - √3)
= 5(5 - √3) + √3(5 - √3)
= 25 - 5√3 + 5√3 - √3√3
= 25 - √3√3
= 25 - 3
the radical √3 was squared and became negative, because it was multiplied by the radical in the conjugate.