Ответы к странице 139

545. Разложите на множители выражение:
1) $15 - x^2$;
2) $49x^2 - 2$;
3) $36p - 64q$, если p ≥ 0, q ≥ 0;
4) c − 100, если c ≥ 0;
5) $a - 8b\sqrt{a} + 16b^2$;
6) $m + 2\sqrt{mn} + n$, если m ≥ 0, n ≥ 0;
7) $a - 4\sqrt{a} + 4$;
8) $5 + \sqrt{5}$;
9) $\sqrt{3p} - p$;
10) $\sqrt{12} + \sqrt{32}$.

Решение:

1) $15 - x^2 = (\sqrt{15})^2 - x^2 = (\sqrt{15} - x)(\sqrt{15} + x)$

2) $49x^2 - 2 = (7x)^2 - \sqrt{2} = (7x - \sqrt{2})(7x + \sqrt{2})$

3) $36p - 64q = (6\sqrt{p})^2 - (8\sqrt{q})^2 = (6\sqrt{p} - 8\sqrt{q})(6\sqrt{p} + 8\sqrt{q})$

4) $c - 100 = (\sqrt{c})^2 - 10^2 = (\sqrt{c} - 10)(\sqrt{c} + 10)$

5) $a - 8b\sqrt{a} + 16b^2 = (\sqrt{a})^2 - 2 * \sqrt{a} * 4b + (4b)^2 = (\sqrt{a} - 4b)^2$

6) $m + 2\sqrt{mn} + n = (\sqrt{m})^2 + 2 * \sqrt{m} * \sqrt{n} + (\sqrt{n})^2 = (\sqrt{m} + \sqrt{n})^2$

7) $a - 4\sqrt{a} + 4 = (\sqrt{a})^2 - 2 * \sqrt{a} * 2 + 2^2 = (\sqrt{a} - 2)^2$

8) $5 + \sqrt{5} = (\sqrt{5})^2 + \sqrt{5} = \sqrt{5}(\sqrt{5} + 1)$

9) $\sqrt{3p} - p = \sqrt{3} * \sqrt{p} - (\sqrt{p})^2 = \sqrt{p}(\sqrt{3} - \sqrt{p})$

10) $\sqrt{12} + \sqrt{32} = \sqrt{4 * 3} + \sqrt{16 * 2} = 2\sqrt{3} + 4\sqrt{2} = 2(\sqrt{3} + 2\sqrt{2})$

546. Сократите дробь:
1) $\frac{a^2 - 7}{a + \sqrt{7}}$;
2) $\frac{\sqrt{3} - b}{3 - b^2}$;
3) $\frac{c - 9}{\sqrt{c} - 3}$;
4) $\frac{a - b}{\sqrt{a} + \sqrt{b}}$;
5) $\frac{5\sqrt{a} - 7\sqrt{b}}{25a - 49b}$;
6) $\frac{100a^2 - 9b}{10a + 3\sqrt{b}}$;
7) $\frac{\sqrt{2} - 1}{\sqrt{6} - \sqrt{3}}$;
8) $\frac{\sqrt{35} + \sqrt{10}}{\sqrt{7} + \sqrt{2}}$;
9) $\frac{\sqrt{15} - \sqrt{6}}{5 - \sqrt{10}}$;
10) $\frac{13 - \sqrt{13}}{\sqrt{13}}$;
11) $\frac{a + 2\sqrt{ab} + b}{\sqrt{a} + \sqrt{b}}$;
12) $\frac{4b^2 - 4b\sqrt{c} + c}{2b - \sqrt{c}}$.

Решение:

1) $\frac{a^2 - 7}{a + \sqrt{7}} = \frac{a^2 - (\sqrt{7})^2}{a + \sqrt{7}} = \frac{(a - \sqrt{7})(a + \sqrt{7})}{a + \sqrt{7}} = a - \sqrt{7}$

2) $\frac{\sqrt{3} - b}{3 - b^2} = \frac{\sqrt{3} - b}{(\sqrt{3})^2 - b^2} = \frac{\sqrt{3} - b}{(\sqrt{3} - b)(\sqrt{3} + b)} = \frac{1}{\sqrt{3} + b}$

3) $\frac{c - 9}{\sqrt{c} - 3} = \frac{(\sqrt{c})^2 - 3^2}{\sqrt{c} - 3} = \frac{(\sqrt{c} - 3)(\sqrt{c} + 3)}{\sqrt{c} - 3} = \sqrt{c} + 3$

4) $\frac{a - b}{\sqrt{a} + \sqrt{b}} = \frac{(\sqrt{a})^2 - (\sqrt{b})^2}{\sqrt{a} + \sqrt{b}} = \frac{(\sqrt{a} - \sqrt{b})(\sqrt{a} + \sqrt{b})}{\sqrt{a} + \sqrt{b}} = \sqrt{a} - \sqrt{b}$

5) $\frac{5\sqrt{a} - 7\sqrt{b}}{25a - 49b} = \frac{5\sqrt{a} - 7\sqrt{b}}{(5\sqrt{a})^2 - (7\sqrt{b})^2} = \frac{5\sqrt{a} - 7\sqrt{b}}{(5\sqrt{a} - 7\sqrt{b})(5\sqrt{a} + 7\sqrt{b})} = \frac{1}{5\sqrt{a} + 7\sqrt{b}}$

6) $\frac{100a^2 - 9b}{10a + 3\sqrt{b}} = \frac{(10a)^2 - (3\sqrt{b})^2}{10a + 3\sqrt{b}} = \frac{(10a - 3\sqrt{b})(10a + 3\sqrt{b})}{10a + 3\sqrt{b}} = 10a - 3\sqrt{b}$

7) $\frac{\sqrt{2} - 1}{\sqrt{6} - \sqrt{3}} = \frac{\sqrt{2} - 1}{\sqrt{2 * 3} - \sqrt{3}} = \frac{\sqrt{2} - 1}{\sqrt{2} * \sqrt{3} - \sqrt{3}} = \frac{\sqrt{2} - 1}{\sqrt{3}(\sqrt{2} - 1)} = \frac{1}{\sqrt{3}}$

8) $\frac{\sqrt{35} + \sqrt{10}}{\sqrt{7} + \sqrt{2}} = \frac{\sqrt{7 * 5} + \sqrt{2 * 5}}{\sqrt{7} + \sqrt{2}} = \frac{\sqrt{7} * \sqrt{5} + \sqrt{2} * \sqrt{5}}{\sqrt{7} + \sqrt{2}} = \frac{\sqrt{5}(\sqrt{7} + \sqrt{2})}{\sqrt{7} + \sqrt{2}} = \sqrt{5}$

9) $\frac{\sqrt{15} - \sqrt{6}}{5 - \sqrt{10}} = \frac{\sqrt{5 * 3} - \sqrt{2 * 3}}{(\sqrt{5})^2 - \sqrt{5 * 2}} = \frac{\sqrt{5} * \sqrt{3} - \sqrt{2} * \sqrt{3}}{(\sqrt{5})^2 - \sqrt{5} * \sqrt{2}} = \frac{\sqrt{3}(\sqrt{5} - \sqrt{2})}{\sqrt{5}(\sqrt{5} - \sqrt{2})} = \frac{\sqrt{3}}{\sqrt{5}}$

10) $\frac{13 - \sqrt{13}}{\sqrt{13}} = \frac{(\sqrt{13})^2 - \sqrt{13}}{\sqrt{13}} = \frac{\sqrt{13}(\sqrt{13} - 1)}{\sqrt{13}} = \sqrt{13} - 1$

11) $\frac{a + 2\sqrt{ab} + b}{\sqrt{a} + \sqrt{b}} = \frac{(\sqrt{a})^2 + 2 * \sqrt{a} * \sqrt{b} + (\sqrt{b})^2}{\sqrt{a} + \sqrt{b}} = \frac{(\sqrt{a} + \sqrt{b})^2}{\sqrt{a} + \sqrt{b}} = \sqrt{a} + \sqrt{b}$

12) $\frac{4b^2 - 4b\sqrt{c} + c}{2b - \sqrt{c}} = \frac{(2b)^2 - 2 * 2b * \sqrt{c} + (\sqrt{c})^2}{2b - \sqrt{c}} = \frac{(2b - \sqrt{c})^2}{2b - \sqrt{c}} = 2b - \sqrt{c}$

547. Сократите дробь:
1) $\frac{x - 25}{\sqrt{x} - 5}$;
2) $\frac{\sqrt{a} + 2}{a - 4}$;
3) $\frac{a - 3}{\sqrt{a} + \sqrt{3}}$;
4) $\frac{\sqrt{10} + \sqrt{5}}{\sqrt{5}}$;
5) $\frac{23 - \sqrt{23}}{\sqrt{23}}$;
6) $\frac{\sqrt{24} - \sqrt{28}}{\sqrt{54} - \sqrt{63}}$;
7) $\frac{\sqrt{a} - \sqrt{b}}{a - 2\sqrt{ab} + b}$;
8) $\frac{b - 8\sqrt{b} + 16}{\sqrt{b} - 4}$.

Решение:

1) $\frac{x - 25}{\sqrt{x} - 5} = \frac{(\sqrt{x})^2 - 5^2}{\sqrt{x} - 5} = \frac{(\sqrt{x} - 5)(\sqrt{x} + 5)}{\sqrt{x} - 5} = \sqrt{x} + 5$

2) $\frac{\sqrt{a} + 2}{a - 4} = \frac{\sqrt{a} + 2}{(\sqrt{a})^2 - 2^2} = \frac{\sqrt{a} + 2}{(\sqrt{a} - 2)(\sqrt{a} + 2)} = \frac{1}{\sqrt{a} - 2}$

3) $\frac{a - 3}{\sqrt{a} + \sqrt{3}} = \frac{(\sqrt{a})^2 - (\sqrt{3})^2}{\sqrt{a} + \sqrt{3}} = \frac{(\sqrt{a} - \sqrt{3})(\sqrt{a} + \sqrt{3})}{\sqrt{a} + \sqrt{3}} = \sqrt{a} - \sqrt{3}$

4) $\frac{\sqrt{10} + \sqrt{5}}{\sqrt{5}} = \frac{\sqrt{2 * 5} + \sqrt{5}}{\sqrt{5}} = \frac{\sqrt{2} * \sqrt{5} + \sqrt{5}}{\sqrt{5}} = \frac{\sqrt{5}(\sqrt{2} + 1)}{\sqrt{5}} = \sqrt{2} + 1$

5) $\frac{23 - \sqrt{23}}{\sqrt{23}} = \frac{(\sqrt{23})^2 - \sqrt{23}}{\sqrt{23}} = \frac{\sqrt{23}(\sqrt{23} - 1)}{\sqrt{23}} = \sqrt{23} - 1$

6) $\frac{\sqrt{24} - \sqrt{28}}{\sqrt{54} - \sqrt{63}} = \frac{\sqrt{4 * 6} - \sqrt{4 * 7}}{\sqrt{9 * 6} - \sqrt{9 * 7}} = \frac{2\sqrt{6} - 2\sqrt{7}}{3\sqrt{6} - 3\sqrt{7}} = \frac{2(\sqrt{6} - \sqrt{7})}{3(\sqrt{6} - \sqrt{7})} = \frac{2}{3}$

7) $\frac{\sqrt{a} - \sqrt{b}}{a - 2\sqrt{ab} + b} = \frac{\sqrt{a} - \sqrt{b}}{(\sqrt{a})^2 - 2 * \sqrt{a} * \sqrt{b} + (\sqrt{b})^2} = \frac{\sqrt{a} - \sqrt{b}}{(\sqrt{a} - \sqrt{b})^2} = \frac{1}{\sqrt{a} - \sqrt{b}}$

8) $\frac{b - 8\sqrt{b} + 16}{\sqrt{b} - 4} = \frac{(\sqrt{b})^2 - 2 * \sqrt{b} * 4 + 4^2}{\sqrt{b} - 4} = \frac{(\sqrt{b} - 4)^2}{\sqrt{b} - 4} = \sqrt{b} - 4$

548. Вынесите множитель из−под знака корня:
1) $\sqrt{3a^2}$, если a ≥ 0;
2) $\sqrt{5b^2}$, если b ≤ 0;
3) $\sqrt{12a^4}$;
4) $\sqrt{c^5}$.

Решение:

1) $\sqrt{3a^2} = \sqrt{3} * \sqrt{a^2} = a\sqrt{3}$, если a ≥ 0

2) $\sqrt{5b^2} = \sqrt{5} * \sqrt{b^2} = -b\sqrt{5}$, если b ≤ 0

3) $\sqrt{12a^4} = \sqrt{12 * (a^2)^2} = \sqrt{12} * \sqrt{(a^2)^2} = \sqrt{4 * 3} * a^2 = 2\sqrt{3} * a^2 = 2a^2\sqrt{3}$

4) $\sqrt{c^5} = \sqrt{c^4 * c} = \sqrt{(c^2)^2 * c} = \sqrt{(c^2)^2} * \sqrt{c} = c^2\sqrt{c}$