Ответы к странице 141
557. Освободитесь от иррациональности в знаменателе дроби:
1) $frac{sqrt{2}}{sqrt{2} + 1}$;
2) $frac{4}{sqrt{7} + sqrt{3}}$;
3) $frac{15}{sqrt{15} — sqrt{12}}$;
4) $frac{19}{2sqrt{5} — 1}$;
5) $frac{1}{sqrt{a} — sqrt{b}}$;
6) $frac{sqrt{3} + 1}{sqrt{3} — 1}$.
Решение:
1) $frac{sqrt{2}}{sqrt{2} + 1} = frac{sqrt{2}(sqrt{2} — 1)}{(sqrt{2} + 1)(sqrt{2} — 1)} = frac{2 — sqrt{2}}{(sqrt{2})^2 — 1^2} = frac{2 — sqrt{2}}{2 — 1} = 2 — sqrt{2}$
2) $frac{4}{sqrt{7} + sqrt{3}} = frac{4(sqrt{7} — sqrt{3})}{(sqrt{7} + sqrt{3})(sqrt{7} — sqrt{3})} = frac{4(sqrt{7} — sqrt{3})}{(sqrt{7})^2 — (sqrt{3})^2} = frac{4(sqrt{7} — sqrt{3})}{7 — 3} = frac{4(sqrt{7} — sqrt{3})}{4} = sqrt{7} — sqrt{3}$
3) $frac{15}{sqrt{15} — sqrt{12}} = frac{15(sqrt{15} + sqrt{12})}{(sqrt{15} — sqrt{12})(sqrt{15} + sqrt{12})} = frac{15(sqrt{15} + sqrt{12})}{(sqrt{15})^2 — (sqrt{12})^2} = frac{15(sqrt{15} + sqrt{12})}{15 — 12} = frac{15(sqrt{15} + sqrt{12})}{3} = 5(sqrt{15} + sqrt{12}) = 5(sqrt{5 * 3} + sqrt{4 * 3}) = 5(sqrt{5} * sqrt{3} + 2sqrt{3}) = 5sqrt{3}(sqrt{5} + 2)$
4) $frac{19}{2sqrt{5} — 1} = frac{19(2sqrt{5} + 1)}{(2sqrt{5} — 1)(2sqrt{5} + 1)} = frac{19(2sqrt{5} + 1)}{(2sqrt{5})^2 — 1^2} = frac{19(2sqrt{5} + 1)}{4 * 5 — 1} = frac{19(2sqrt{5} + 1)}{20 — 1} = frac{19(2sqrt{5} + 1)}{19} = 2sqrt{5} + 1$
5) $frac{1}{sqrt{a} — sqrt{b}} = frac{1 * (sqrt{a} + sqrt{b})}{(sqrt{a} — sqrt{b})(sqrt{a} + sqrt{b})} = frac{sqrt{a} + sqrt{b}}{(sqrt{a})^2 — (sqrt{b})^2} = frac{sqrt{a} + sqrt{b}}{a — b}$
6) $frac{sqrt{3} + 1}{sqrt{3} — 1} = frac{(sqrt{3} + 1)(sqrt{3} + 1)}{(sqrt{3} — 1)(sqrt{3} + 1)} = frac{(sqrt{3} + 1)^2}{(sqrt{3})^2 — 1^2} = frac{(sqrt{3})^2 + 2 * sqrt{3} * 1 + 1^2}{3 — 1} = frac{3 + 2sqrt{3} + 1}{2} = frac{4 + 2sqrt{3}}{2} = frac{2(2 + sqrt{3})}{2} = 2 + sqrt{3}$
558. Освободитесь от иррациональности в знаменателе дроби:
1) $frac{sqrt{5}}{sqrt{5} — 2}$;
2) $frac{8}{sqrt{10} — sqrt{2}}$;
3) $frac{9}{sqrt{x} + sqrt{y}}$;
4) $frac{2 — sqrt{2}}{2 + sqrt{2}}$.
Решение:
1) $frac{sqrt{5}}{sqrt{5} — 2} = frac{sqrt{5}(sqrt{5} + 2)}{(sqrt{5} — 2)(sqrt{5} + 2)} = frac{sqrt{5}(sqrt{5} + 2)}{(sqrt{5})^2 — 2^2} = frac{sqrt{5}(sqrt{5} + 2)}{5 — 4} = frac{5 + 2sqrt{5}}{1} = 5 + 2sqrt{5}$
2) $frac{8}{sqrt{10} — sqrt{2}} = frac{8(sqrt{10} + sqrt{2})}{(sqrt{10} — sqrt{2})(sqrt{10} + sqrt{2})} = frac{8(sqrt{10} + sqrt{2})}{(sqrt{10})^2 — (sqrt{2})^2} = frac{8(sqrt{10} + sqrt{2})}{10 — 2} = frac{8(sqrt{10} + sqrt{2})}{8} = sqrt{10} + sqrt{2}$
3) $frac{9}{sqrt{x} + sqrt{y}} = frac{9(sqrt{x} — sqrt{y})}{(sqrt{x} + sqrt{y})(sqrt{x} — sqrt{y})} = frac{9(sqrt{x} — sqrt{y})}{(sqrt{x})^2 — (sqrt{y})^2} = frac{9(sqrt{x} — sqrt{y})}{x — y}$
4) $frac{2 — sqrt{2}}{2 + sqrt{2}} = frac{(2 — sqrt{2})(2 — sqrt{2})}{(2 + sqrt{2})(2 — sqrt{2})} = frac{(2 — sqrt{2})^2}{2^2 — (sqrt{2})^2} = frac{2^2 — 2 * 2sqrt{2} + (sqrt{2})^2}{4 — 2} = frac{4 — 4sqrt{2} + 2}{2} = frac{6 — 4sqrt{2}}{2} = frac{2(3 — 2sqrt{2})}{2} = 3 — 2sqrt{2}$
559. Докажите равенство:
1) $frac{1}{5 — 2sqrt{6}} + frac{1}{5 + 2sqrt{6}} = 10$;
2) $frac{2}{3sqrt{2} + 4} — frac{2}{3sqrt{2} — 4} = -8$;
3) $frac{sqrt{2} + 1}{sqrt{2} — 1} — frac{sqrt{2} — 1}{sqrt{2} + 1} = 4sqrt{2}$.
Решение:
1) $frac{1}{5 — 2sqrt{6}} + frac{1}{5 + 2sqrt{6}} = frac{5 + 2sqrt{6} + 5 — 2sqrt{6}}{(5 — 2sqrt{6})(5 + 2sqrt{6})} = frac{10}{5^2 — (2sqrt{6})^2)} = frac{10}{25 — 4 * 6} = frac{10}{25 — 24} = frac{10}{1} = 10$
2) $frac{2}{3sqrt{2} + 4} — frac{2}{3sqrt{2} — 4} = frac{2(3sqrt{2} — 4) — 2(3sqrt{2} + 4)}{(3sqrt{2} + 4)(3sqrt{2} — 4)} = frac{6sqrt{2} — 8 — 6sqrt{2} — 8}{(3sqrt{2})^2 — 4^2} = frac{-16}{9 * 2 — 16} = frac{-16}{18 — 16} = frac{-16}{2} = -8$
3) $frac{sqrt{2} + 1}{sqrt{2} — 1} — frac{sqrt{2} — 1}{sqrt{2} + 1} = frac{(sqrt{2} + 1)^2 — (sqrt{2} — 1)^2}{(sqrt{2} — 1)(sqrt{2} + 1)} = frac{2 + 2sqrt{2} + 1 — (2 — 2sqrt{2} + 1)}{(sqrt{2})^2 — 1^2} = frac{3 + 2sqrt{2} — 2 + 2sqrt{2} — 1}{2 — 1} = frac{4sqrt{2}}{1} = 4sqrt{2}$
560. Докажите, что значением выражения является рациональное число:
1) $frac{6}{3 + 2sqrt{3}} + frac{6}{3 — 2sqrt{3}}$;
2) $frac{sqrt{11} + sqrt{6}}{sqrt{11} — sqrt{6}} + frac{sqrt{11} — sqrt{6}}{sqrt{11} + sqrt{6}}$.
Решение:
1) $frac{6}{3 + 2sqrt{3}} + frac{6}{3 — 2sqrt{3}} = frac{6(3 — 2sqrt{3}) + 6(3 + 2sqrt{3})}{(3 + 2sqrt{3})(3 — 2sqrt{3})} = frac{18 — 12sqrt{3} + 18 + 12sqrt{3}}{3^2 — (2sqrt{3})^2} = frac{36}{9 — 4 * 3} = frac{36}{9 — 12} = frac{36}{-3} = -12$ − рациональное число
2) $frac{sqrt{11} + sqrt{6}}{sqrt{11} — sqrt{6}} + frac{sqrt{11} — sqrt{6}}{sqrt{11} + sqrt{6}} = frac{(sqrt{11} + sqrt{6})^2 + (sqrt{11} — sqrt{6})^2}{(sqrt{11} — sqrt{6})(sqrt{11} + sqrt{6})} = frac{(sqrt{11})^2 + 2 * sqrt{11} * sqrt{6} + (sqrt{6})^2 + (sqrt{11})^2 — 2 * sqrt{11} * sqrt{6} + (sqrt{6})^2}{(sqrt{11})^2 — (sqrt{6})^2} = frac{11 + 2sqrt{66} + 6 + 11 — 2sqrt{66} + 6}{11 — 6} = frac{34}{5} = 6frac{4}{5}$ − рациональное число
561. Упростите выражение:
1) $frac{a}{sqrt{a} — 2} — frac{4sqrt{a} — 4}{sqrt{a} — 2}$;
2) $frac{sqrt{m} + 1}{sqrt{m} — 2} — frac{sqrt{m} + 3}{sqrt{m}}$;
3) $frac{sqrt{y} + 4}{sqrt{xy} + y} — frac{sqrt{x} — 4}{x + sqrt{xy}}$;
4) $frac{sqrt{a}}{sqrt{a} + 4} — frac{a}{a — 16}$;
5) $frac{a}{sqrt{ab} — b} + frac{sqrt{b}}{sqrt{b} — sqrt{a}}$;
6) $frac{a + sqrt{a}}{sqrt{b}} * frac{b}{2sqrt{a} + 2}$;
7) $frac{sqrt{c} — 5}{sqrt{c}} : frac{с — 25}{3с}$;
8) $(sqrt{a} — frac{a}{sqrt{a} + 1}) : frac{sqrt{a}}{a — 1}$;
9) $(frac{sqrt{a} + sqrt{b}}{sqrt{b}} + frac{sqrt{b}}{sqrt{a} — sqrt{b}}) : frac{sqrt{a}}{sqrt{b}}$;
10) $(frac{sqrt{x} — 3}{sqrt{x} + 3} + frac{12sqrt{x}}{x — 9}) : frac{sqrt{x} + 3}{x — 3sqrt{x}}$.
Решение:
1) $frac{a}{sqrt{a} — 2} — frac{4sqrt{a} — 4}{sqrt{a} — 2} = frac{a — (4sqrt{a} — 4)}{sqrt{a} — 2} = frac{a — 4sqrt{a} + 4}{sqrt{a} — 2} = frac{(sqrt{a})^2 — 2 * 2sqrt{a} + 2^2}{sqrt{a} — 2} = frac{(sqrt{a} — 2)^2}{sqrt{a} — 2} = sqrt{a} — 2$
2) $frac{sqrt{m} + 1}{sqrt{m} — 2} — frac{sqrt{m} + 3}{sqrt{m}} = frac{sqrt{m}(sqrt{m} + 1) — (sqrt{m} + 3)(sqrt{m} — 2)}{sqrt{m}(sqrt{m} — 2)} = frac{m + sqrt{m} — (m + 3sqrt{m} — 2sqrt{m} — 6)}{sqrt{m}(sqrt{m} — 2)} = frac{m + sqrt{m} — m — 3sqrt{m} + 2sqrt{m} + 6}{sqrt{m}(sqrt{m} — 2)} = frac{6}{m — 2sqrt{m}}$
3) $frac{sqrt{y} + 4}{sqrt{xy} + y} — frac{sqrt{x} — 4}{x + sqrt{xy}} = frac{sqrt{y} + 4}{sqrt{x} * sqrt{y} + (sqrt{y})^2} — frac{sqrt{x} — 4}{(sqrt{x})^2 + sqrt{x} * sqrt{y}} = frac{sqrt{y} + 4}{sqrt{y}(sqrt{x} + sqrt{y})} — frac{sqrt{x} — 4}{sqrt{x}(sqrt{x} + sqrt{y})} = frac{sqrt{x}(sqrt{y} + 4) — sqrt{y}(sqrt{x} — 4)}{sqrt{xy}(sqrt{x} + sqrt{y})} = frac{sqrt{xy} + 4sqrt{x} — sqrt{xy} + 4sqrt{y}}{sqrt{xy}(sqrt{x} + sqrt{y})} = frac{4sqrt{x} + 4sqrt{y}}{sqrt{xy}(sqrt{x} + sqrt{y})} = frac{4(sqrt{x} + sqrt{y})}{sqrt{xy}(sqrt{x} + sqrt{y})} = frac{4}{sqrt{xy}}$
4) $frac{sqrt{a}}{sqrt{a} + 4} — frac{a}{a — 16} = frac{sqrt{a}}{sqrt{a} + 4} — frac{a}{(sqrt{a})^2 — 4^2} = frac{sqrt{a}}{sqrt{a} + 4} — frac{a}{(sqrt{a} — 4)(sqrt{a} + 4)} = frac{sqrt{a}(sqrt{a} — 4) — a}{(sqrt{a} — 4)(sqrt{a} + 4)} = frac{a — 4sqrt{a} — a}{a — 16} = frac{-4sqrt{a}}{a — 16} = -frac{4sqrt{a}}{a — 16}$
5) $frac{a}{sqrt{ab} — b} + frac{sqrt{b}}{sqrt{b} — sqrt{a}} = frac{a}{sqrt{b} * sqrt{a} — (sqrt{b})^2} + frac{sqrt{b}}{sqrt{b} — sqrt{a}} = frac{a}{sqrt{b}(sqrt{a} — sqrt{b})} + frac{sqrt{b}}{sqrt{b} — sqrt{a}} = frac{a}{sqrt{b}(sqrt{a} — sqrt{b})} — frac{sqrt{b}}{sqrt{a} — sqrt{b}} = frac{a — (sqrt{b})^2}{sqrt{b}(sqrt{a} — sqrt{b})} = frac{(sqrt{a})^2 — (sqrt{b})^2}{sqrt{b}(sqrt{a} — sqrt{b})} = frac{(sqrt{a} — sqrt{b})(sqrt{a} + sqrt{b})}{sqrt{b}(sqrt{a} — sqrt{b})} = frac{sqrt{a} + sqrt{b}}{sqrt{b}}$
6) $frac{a + sqrt{a}}{sqrt{b}} * frac{b}{2sqrt{a} + 2} = frac{(sqrt{a})^2 + sqrt{a}}{sqrt{b}} * frac{(sqrt{b})^2}{2(sqrt{a} + 1)} = frac{sqrt{a}(sqrt{a} + 1)}{1} * frac{sqrt{b}}{2(sqrt{a} + 1)} = frac{sqrt{a}}{1} * frac{sqrt{b}}{2} = frac{sqrt{ab}}{2}$
7) $frac{sqrt{c} — 5}{sqrt{c}} : frac{с — 25}{3с} = frac{sqrt{c} — 5}{sqrt{c}} * frac{3с}{с — 25} = frac{sqrt{c} — 5}{sqrt{c}} * frac{3 * (sqrt{c})^2}{(sqrt{c})^2 — 5^2} = frac{sqrt{c} — 5}{1} * frac{3sqrt{c}}{(sqrt{c} — 5)(sqrt{c} + 5)} = frac{3sqrt{c}}{sqrt{c} + 5}$
8) $(sqrt{a} — frac{a}{sqrt{a} + 1}) : frac{sqrt{a}}{a — 1} = frac{sqrt{a}(sqrt{a} + 1) — a}{sqrt{a} + 1} * frac{(sqrt{a})^2 — 1^2}{sqrt{a}} = frac{a + sqrt{a} — a}{sqrt{a} + 1} * frac{(sqrt{a} — 1)(sqrt{a} + 1)}{sqrt{a}} = frac{sqrt{a}}{1} * frac{sqrt{a} — 1}{sqrt{a}} = sqrt{a} — 1$
9) $(frac{sqrt{a} + sqrt{b}}{sqrt{b}} + frac{sqrt{b}}{sqrt{a} — sqrt{b}}) : frac{sqrt{a}}{sqrt{b}} = frac{(sqrt{a} — sqrt{b})(sqrt{a} + sqrt{b}) + (sqrt{b})^2}{sqrt{b}(sqrt{a} — sqrt{b})} * frac{sqrt{b}}{sqrt{a}} = frac{a — b + b}{sqrt{a} — sqrt{b}} * frac{1}{sqrt{a}} = frac{a}{sqrt{a} — sqrt{b}} * frac{1}{sqrt{a}} = frac{(sqrt{a})^2}{sqrt{a} — sqrt{b}} * frac{1}{sqrt{a}} = frac{sqrt{a}}{sqrt{a} — sqrt{b}}$
10) $(frac{sqrt{x} — 3}{sqrt{x} + 3} + frac{12sqrt{x}}{x — 9}) : frac{sqrt{x} + 3}{x — 3sqrt{x}} = (frac{sqrt{x} — 3}{sqrt{x} + 3} + frac{12sqrt{x}}{(sqrt{x})^2 — 3^2}) * frac{x — 3sqrt{x}}{sqrt{x} + 3} = frac{(sqrt{x} — 3)^2 + 12sqrt{x}}{(sqrt{x} — 3)(sqrt{x} + 3)} * frac{(sqrt{x})^2 — 3sqrt{x}}{sqrt{x} + 3} = frac{(sqrt{x})^2 — 2 * 3sqrt{x} + 3^2 + 12sqrt{x}}{(sqrt{x} — 3)(sqrt{x} + 3)} * frac{sqrt{x}(sqrt{x} — 3)}{sqrt{x} + 3} = frac{x — 6sqrt{x} + 9 + 12sqrt{x}}{sqrt{x} + 3} * frac{sqrt{x}}{sqrt{x} + 3} = frac{x + 6sqrt{x} + 9}{sqrt{x} + 3} * frac{sqrt{x}}{sqrt{x} + 3} = frac{(sqrt{x})^2 + 2 * 3sqrt{x} + 3^2}{sqrt{x} + 3} * frac{sqrt{x}}{sqrt{x} + 3} = frac{(sqrt{x} + 3)^2}{sqrt{x} + 3} * frac{sqrt{x}}{sqrt{x} + 3} = sqrt{x}$
562. Упростите выражение:
1) $frac{sqrt{a} — 3}{sqrt{a} + 1} — frac{sqrt{a} — 4}{sqrt{a}}$;
2) $frac{sqrt{a} + 1}{a — sqrt{ab}} — frac{sqrt{b} + 1}{sqrt{ab} — b}$;
3) $frac{sqrt{x}}{y — 2sqrt{y}} : frac{sqrt{x}}{3sqrt{y} — 6}$;
4) $frac{sqrt{m}}{sqrt{m} — sqrt{n}} : (frac{sqrt{m} + sqrt{n}}{sqrt{n}} + frac{sqrt{n}}{sqrt{m} — sqrt{n}})$;
5) $(frac{sqrt{x} + 1}{sqrt{x} — 1} — frac{4sqrt{x}}{x — 1}) * frac{x + sqrt{x}}{sqrt{x} — 1}$;
6) $frac{a — 64}{sqrt{a} + 3} * frac{1}{a + 8sqrt{a}} — frac{sqrt{a} + 8}{a — 3sqrt{a}}$.
Решение:
1) $frac{sqrt{a} — 3}{sqrt{a} + 1} — frac{sqrt{a} — 4}{sqrt{a}} = frac{sqrt{a}(sqrt{a} — 3) — (sqrt{a} + 1)(sqrt{a} — 4)}{sqrt{a}(sqrt{a} + 1)} = frac{a — 3sqrt{a} — (a + sqrt{a} — 4sqrt{a} — 4)}{sqrt{a}(sqrt{a} + 1)} = frac{a — 3sqrt{a} — a — sqrt{a} + 4sqrt{a} + 4}{sqrt{a}(sqrt{a} + 1)} = frac{4}{a + sqrt{a}}$
2) $frac{sqrt{a} + 1}{a — sqrt{ab}} — frac{sqrt{b} + 1}{sqrt{ab} — b} = frac{sqrt{a} + 1}{sqrt{a}(sqrt{a} — sqrt{b})} — frac{sqrt{b} + 1}{sqrt{b}(sqrt{a} — sqrt{b})} = frac{sqrt{b}(sqrt{a} + 1) — sqrt{a}(sqrt{b} + 1)}{sqrt{ab}(sqrt{a} — sqrt{b})} = frac{sqrt{ab} + sqrt{b} — sqrt{ab} — sqrt{a}}{sqrt{ab}(sqrt{a} — sqrt{b})} = frac{sqrt{b} — sqrt{a}}{sqrt{ab}(sqrt{a} — sqrt{b})} = -frac{sqrt{a} — sqrt{b}}{sqrt{ab}(sqrt{a} — sqrt{b})} = -frac{1}{sqrt{ab}} = -sqrt{frac{1}{ab}}$
3) $frac{sqrt{x}}{y — 2sqrt{y}} : frac{sqrt{x}}{3sqrt{y} — 6} = frac{sqrt{x}}{sqrt{y}(sqrt{y} — 2)} : frac{sqrt{x}}{3(sqrt{y} — 2)} = frac{sqrt{x}}{sqrt{y}(sqrt{y} — 2)} * frac{3(sqrt{y} — 2)}{sqrt{x}} = frac{1}{sqrt{y}} * frac{3}{1} = frac{3}{sqrt{y}}$
4) $frac{sqrt{m}}{sqrt{m} — sqrt{n}} : (frac{sqrt{m} + sqrt{n}}{sqrt{n}} + frac{sqrt{n}}{sqrt{m} — sqrt{n}}) = frac{sqrt{m}}{sqrt{m} — sqrt{n}} : frac{(sqrt{m} — sqrt{n})(sqrt{m} + sqrt{n}) + (sqrt{n})^2}{sqrt{n}(sqrt{m} — sqrt{n})} = frac{sqrt{m}}{sqrt{m} — sqrt{n}} : frac{m — n + n}{sqrt{n}(sqrt{m} — sqrt{n})} = frac{sqrt{m}}{sqrt{m} — sqrt{n}} : frac{m}{sqrt{n}(sqrt{m} — sqrt{n})} = frac{sqrt{m}}{sqrt{m} — sqrt{n}} * frac{sqrt{n}(sqrt{m} — sqrt{n})}{m} = frac{sqrt{n}}{sqrt{m}} = sqrt{frac{n}{m}}$
5) $(frac{sqrt{x} + 1}{sqrt{x} — 1} — frac{4sqrt{x}}{x — 1}) * frac{x + sqrt{x}}{sqrt{x} — 1} = (frac{sqrt{x} + 1}{sqrt{x} — 1} — frac{4sqrt{x}}{(sqrt{x} — 1)(sqrt{x} + 1)}) * frac{sqrt{x}(sqrt{x} + 1)}{sqrt{x} — 1} = frac{(sqrt{x} + 1)^2 — 4sqrt{x}}{(sqrt{x} — 1)(sqrt{x} + 1)} * frac{sqrt{x}(sqrt{x} + 1)}{sqrt{x} — 1} = frac{x + 2sqrt{x} + 1 — 4sqrt{x}}{sqrt{x} — 1} * frac{sqrt{x}}{sqrt{x} — 1} = frac{x — 2sqrt{x} + 1}{sqrt{x} — 1} * frac{sqrt{x}}{sqrt{x} — 1} = frac{sqrt{x}(sqrt{x} — 1)^2}{(sqrt{x} — 1)^2} = sqrt{x}$
6) $frac{a — 64}{sqrt{a} + 3} * frac{1}{a + 8sqrt{a}} — frac{sqrt{a} + 8}{a — 3sqrt{a}} = frac{(sqrt{a} — 8)(sqrt{a} + 8)}{sqrt{a} + 3} * frac{1}{sqrt{a}(sqrt{a} + 8)} — frac{sqrt{a} + 8}{sqrt{a}(sqrt{a} — 3)} = frac{sqrt{a} — 8}{sqrt{a} + 3} * frac{1}{sqrt{a}} — frac{sqrt{a} + 8}{sqrt{a}(sqrt{a} — 3)} = frac{sqrt{a} — 8}{sqrt{a}(sqrt{a} + 3)} — frac{sqrt{a} + 8}{sqrt{a}(sqrt{a} — 3)} = frac{(sqrt{a} — 8)(sqrt{a} — 3) — (sqrt{a} + 8)(sqrt{a} + 3)}{sqrt{a}(sqrt{a} — 3)(sqrt{a} + 3)} = frac{a — 8sqrt{a} — 3sqrt{a} + 24 — (a + 8sqrt{a} + 3sqrt{a} + 24)}{sqrt{a}(sqrt{a} — 3)(sqrt{a} + 3)} = frac{a — 8sqrt{a} — 3sqrt{a} + 24 — a — 8sqrt{a} — 3sqrt{a} — 24}{sqrt{a}(sqrt{a} — 3)(sqrt{a} + 3)} = frac{-22sqrt{a}}{sqrt{a}(a — 9)} = -frac{22}{a — 9}$