01. Write the expression as a simplified rational number.
(a) (3)/(50) + (7)/(30)
(b) (8)/(63) + (5)/(42)
(c) (5)/(24) - (3)/(20)
(d) (7)/(54) - (5)/(72)
02. Simplify the expression.
(a) (2x^(2) + 7x + 3)/(2x^(2) - 7x - 4)
(b) (2x^(2) + 7x - 15)/(3x^(2) + 17x + 10)
(c) (y^(2) - 25)/(y^(3) - 125)
(d) (y^(2) - 9)/(y^(3) + 27)
(e) (12 + r - r^(2))/(r^(3) + 3r^(2))
(f) (9x^(2) - 4)/(3x^(2) - 5x + 2) * (9x^(4) - 6x^(3) + 4x^(2))/(27x^(4) + 8x)
(g) (4x^(2) - 9)/(2x^(2) + 7x + 6) * (4x^(4) + 6x^(3) + 9x^(2))/(8x^(7) - 27x^(4)
(h) (5a^(2) + 12a + 4)/(a^(4) - 16) -:(25a^(2) + 20a + 4)/(a^(2) - 2a)
(i) (a^(3) - 8)/(a^(2) - 4) -:(a^(3))/(a^(3) + 8)
(j) (6)/(x^(2) - 4) - (3x)/(x^(2) - 4)
(k) (4)/(3s + 1) - (11)/((3s + 1)^(2))
(l) (2)/(x) + (3x + 1)/(x^(2)) - (x - 2)/(x^(3))
(m) (3t)/(t + 2) + (5t)/(t - 2) - (40)/(t^(2) - 4)
(n) (4x)/(3x - 4) + (8)/(3x^(2) - 4x) + (2)/(x)
(o) (p^(4) + 3p^(3) - 8p - 24)/(p^(3) - 2p^(2) - 9p + 18)
(p) (2ac + bc - 6ad - 3bd)/(6ac + 2ad + 3bc + bd)
(q) 3 + (5)/(u) + (2u)/(3u + 1)
(r) ((b)/(a) - (a)/(b))/((1)/(a) - (1)/(b))
(s) ((1)/(x + 2) - 5)/((4)/(x) - x)
(t) (y^(-1) + x^(-1))/((xy)^(-1))
(u) ((5)/(x - 1) - (5)/(a - 1))/(x - a)
(v) ((x + h)^(2) - 3(x + h) - (x^(2) - 3x))/(h)
(w) ((1)/(x + h) - (1)/(x))/(h)
(x) ((4)/(3x + 3h - 1) - (4)/(3x - 1))/(h)
03. Rationalize the denominator.
(a) (sqrt(t) + 5)/(sqrt(t) - 5)
(b) (sqrt(t) - 7)/(sqrt(t) + 7)
(c) (81x^(2) - 16y^(2))/(3sqrt(x) - 2sqrt(y))
(d) (16x^(2) - y^(2))/(2sqrt(x) - sqrt(y))
04. Rationalize the numerator.
(a) (sqrt(a) - sqrt(b))/(a^(2) - b^(2))
(b) (sqrt(b) + sqrt(c))/(b^(2) - c^(2))
(c) (sqrt(2(x + h) + 1) - sqrt(2x + 1))/(h)
(d) (sqrt(1 - x - h) - sqrt(1 - x))/(h)
05. Express as a sum of terms of the form ax^(r), where r is a rational number.
(a) (3x^(2) - x + 7)/(x^((2)/(3)))
(b) (x^(2) + 4x - 6)/(sqrt(x))
(c) ((x^(2) + 2)^(2))/(x^(5))
(d) ((sqrt(x) - 3)^(2))/(x^(3))