Ответы к странице 130

498. Вычислите значение выражения:
1) $\sqrt{9 * 25}$;
2) $\sqrt{16 * 2500}$;
3) $\sqrt{0,64 * 36}$;
4) $\sqrt{400 * 1,44}$;
5) $\sqrt{0,09 * 0,04}$;
6) $\sqrt{6,25 * 0,16}$;
7) $\sqrt{6^2 * 3^4}$;
8) $\sqrt{7^2 * 2^8}$;
9) $\sqrt{25 * 64 * 0,36}$;
10) $\sqrt{0,01 * 0,81 * 2500}$;
11) $\sqrt{\frac{81}{100}}$;
12) $\sqrt{\frac{49}{256}}$;
13) $\sqrt{3\frac{13}{36}}$;
14) $\sqrt{3\frac{1}{16} * 2\frac{14}{25}}$;
15) $\sqrt{\frac{169}{36 * 81}}$;
16) $\sqrt{\frac{121 * 256}{25 * 100}}$.

Решение:

1) $\sqrt{9 * 25} = \sqrt{9} * \sqrt{25} = 3 * 5 = 15$

2) $\sqrt{16 * 2500} = \sqrt{16} * \sqrt{2500} = 4 * 50 = 200$

3) $\sqrt{0,64 * 36} = \sqrt{0,64} * \sqrt{36} = 0,8 * 6 = 4,8$

4) $\sqrt{400 * 1,44} = \sqrt{400} * \sqrt{1,44} = 20 * 1,2 = 24$

5) $\sqrt{0,09 * 0,04} = \sqrt{0,09} * \sqrt{0,04} = 0,3 * 0,2 = 0,06$

6) $\sqrt{6,25 * 0,16} = \sqrt{6,25} * \sqrt{0,16} = 2,5 * 0,4 = 1$

7) $\sqrt{6^2 * 3^4} = \sqrt{6^2} * \sqrt{3^4} = \sqrt{6^2} * \sqrt{(3^2)^2} = |6| * |3^2| = 6 * 9 = 54$

8) $\sqrt{7^2 * 2^8} = \sqrt{7^2} * \sqrt{(2^4)^2} = |7| * |2^4| = 7 * 16 = 112$

9) $\sqrt{25 * 64 * 0,36} = \sqrt{25} * \sqrt{64} * \sqrt{0,36} = 5 * 8 * 0,6 = 40 * 0,6 = 24$

10) $\sqrt{0,01 * 0,81 * 2500} = \sqrt{0,01} * \sqrt{0,81} * \sqrt{2500} = 0,1 * 0,9 * 50 = 0,9 * 5 = 4,5$

11) $\sqrt{\frac{81}{100}} = \frac{\sqrt{81}}{\sqrt{100}} = \frac{9}{10} = 0,9$

12) $\sqrt{\frac{49}{256}} = \frac{\sqrt{49}}{\sqrt{256}} = \frac{7}{16}$

13) $\sqrt{3\frac{13}{36}} = \sqrt{\frac{121}{36}} = \frac{\sqrt{121}}{\sqrt{36}} = \frac{11}{6} = 1\frac{5}{6}$

14) $\sqrt{3\frac{1}{16} * 2\frac{14}{25}} = \sqrt{\frac{49}{16} * \frac{64}{25}} = \sqrt{\frac{49}{16}} * \sqrt{\frac{64}{25}} = \frac{\sqrt{49}}{\sqrt{16}} * \frac{\sqrt{64}}{\sqrt{25}} = \frac{7}{4} * \frac{8}{5} = \frac{7}{1} * \frac{2}{5} = \frac{14}{5} = 2\frac{4}{5}$

15) $\sqrt{\frac{169}{36 * 81}} = \frac{\sqrt{169}}{\sqrt{36} * \sqrt{81}} = \frac{\sqrt{169}}{\sqrt{36 * 81}} = \frac{13}{6 * 9} = \frac{13}{54}$

16) $\sqrt{\frac{121 * 256}{25 * 100}} = \frac{\sqrt{121 * 256}}{\sqrt{25 * 100}} = \frac{\sqrt{121} * \sqrt{256}}{\sqrt{25} * \sqrt{100}} = \frac{11 * 16}{5 * 10} = \frac{11 * 8}{5 * 5} = \frac{88}{25} = 3\frac{13}{25} $

499. Чему равно значение выражения:
1) $\sqrt{36 * 81}$;
2) $\sqrt{900 * 49}$;
3) $\sqrt{16 * 0,25}$;
4) $\sqrt{9 * 1,69}$;
5) $\sqrt{0,36 * 1,21}$;
6) $\sqrt{5^2 * 3^6}$;
7) $\sqrt{4^4 * 3^2}$;
8) $\sqrt{2^6 * 5^2}$;
9) $\sqrt{2,25 * 0,04 * 1600}$;
10) $\sqrt{13\frac{4}{9}}$;
11) $\sqrt{1\frac{7}{9} * \frac{4}{25}}$;
12) $\sqrt{\frac{1}{16} * \frac{9}{25}}$?

Решение:

1) $\sqrt{36 * 81} = \sqrt{36} * \sqrt{81} = 6 * 9 = 54$

2) $\sqrt{900 * 49} = \sqrt{900} * \sqrt{49} = 30 * 7 = 210$

3) $\sqrt{16 * 0,25} = \sqrt{16} * \sqrt{0,25} = 4 * 0,5 = 2$

4) $\sqrt{9 * 1,69} = \sqrt{9} * \sqrt{1,69} = 3 * 1,3 = 3,9$

5) $\sqrt{0,36 * 1,21} = \sqrt{0,36} * \sqrt{1,21} = 0,6 * 1,1 = 0,66$

6) $\sqrt{5^2 * 3^6} = \sqrt{5^2} * \sqrt{3^6} = \sqrt{5^2} * \sqrt{(3^3)^2} = |5| * |3^3| = 5 * |27| = 5 * 27 = 135$

7) $\sqrt{4^4 * 3^2} = \sqrt{4^4} * \sqrt{3^2} = \sqrt{(4^2)^2} * \sqrt{3^2} = |4^2| * |3| = |16| * 3 = 16 * 3 = 48$

8) $\sqrt{2^6 * 5^2} = \sqrt{2^6} * \sqrt{5^2} = \sqrt{(2^3)^2} * \sqrt{5^2} = |2^3| * |5| = |8| * 5 = 8 * 5 = 40$

9) $\sqrt{2,25 * 0,04 * 1600} = \sqrt{2,25} * \sqrt{0,04} * \sqrt{1600} = 1,5 * 0,2 * 40 = 0,3 * 40 = 12$

10) $\sqrt{13\frac{4}{9}} = \sqrt{\frac{121}{9}} = \frac{\sqrt{121}}{\sqrt{9}} = \frac{11}{3} = 3\frac{2}{3}$

11) $\sqrt{1\frac{7}{9} * \frac{4}{25}} = \sqrt{\frac{16}{9}} * \sqrt{\frac{4}{25}} = \frac{\sqrt{16}}{\sqrt{9}} * \frac{\sqrt{4}}{\sqrt{25}} = \frac{4}{3} * \frac{2}{5} = \frac{8}{15}$

12) $\sqrt{\frac{1}{16} * \frac{9}{25}} = \sqrt{\frac{1}{16}} * \sqrt{\frac{9}{25}} = \frac{\sqrt{1}}{\sqrt{16}} * \frac{\sqrt{9}}{\sqrt{25}} = \frac{1}{4} * \frac{3}{5} = \frac{3}{20}$

500. Найдите значение выражения:
1) $\sqrt{12} * \sqrt{3}$;
2) $\sqrt{32} * \sqrt{2}$;
3) $\sqrt{18} * \sqrt{50}$;
4) $\sqrt{0,009} * \sqrt{1000}$;
5) $\sqrt{200} * \sqrt{0,18}$;
6) $\sqrt{13} * \sqrt{2} * \sqrt{26}$;
7) $\sqrt{2,4} * \sqrt{1\frac{2}{3}}$;
8) $\sqrt{\frac{2}{11}} * \sqrt{8} * \sqrt{\frac{1}{11}}$;
9) $\sqrt{2^3 * 3} * \sqrt{2^5 * 3^3}$.

Решение:

1) $\sqrt{12} * \sqrt{3} = \sqrt{12 * 3} = \sqrt{36} = 6$

2) $\sqrt{32} * \sqrt{2} = \sqrt{32 * 2} = \sqrt{64} = 8$

3) $\sqrt{18} * \sqrt{50} = \sqrt{18 * 50} = \sqrt{900} = 30$

4) $\sqrt{0,009} * \sqrt{1000} = \sqrt{0,009 * 1000} = \sqrt{9} = 3$

5) $\sqrt{200} * \sqrt{0,18} = \sqrt{200 * 0,18} = \sqrt{36} = 6$

6) $\sqrt{13} * \sqrt{2} * \sqrt{26} = \sqrt{13 * 2 * 26} = \sqrt{26 * 26} = \sqrt{26^2} = |26| = 26$

7) $\sqrt{2,4} * \sqrt{1\frac{2}{3}} = \sqrt{2,4 * 1\frac{2}{3}} = \sqrt{\frac{24}{10} * \frac{5}{3}} = \sqrt{\frac{12}{5} * \frac{5}{3}} = \sqrt{\frac{4}{1} * \frac{1}{1}} = \sqrt{4} = 2$

8) $\sqrt{\frac{2}{11}} * \sqrt{8} * \sqrt{\frac{1}{11}} = \sqrt{\frac{2}{11} * 8 * \frac{1}{11}} = \sqrt{\frac{16}{11^2}} = \frac{4}{|11|} = \frac{4}{11}$

9) $\sqrt{2^3 * 3} * \sqrt{2^5 * 3^3} = \sqrt{2^3 * 3 * 2^5 * 3^3} = \sqrt{2^{3 + 5} * 3^{1 + 3}} = \sqrt{2^{8} * 3^{4}} = \sqrt{2^{8}} * \sqrt{3^{4}} = \sqrt{(2^4)^2} * \sqrt{(3^2)^2} = |2^4| * |3^2| = |16| * |9| = 144$

501. Найдите значение выражения:
1) $\sqrt{27} * \sqrt{3}$;
2) $\sqrt{18} * \sqrt{2}$;
3) $\sqrt{10} * \sqrt{12,1}$;
4) $\sqrt{0,5} * \sqrt{50}$;
5) $\sqrt{1\frac{3}{7}} * \sqrt{2,8}$;
6) $\sqrt{5 * 2^3} * \sqrt{5^3 * 2^3}$.

Решение:

1) $\sqrt{27} * \sqrt{3} = \sqrt{27 * 3} = \sqrt{81} = 9$

2) $\sqrt{18} * \sqrt{2} = \sqrt{18 * 2} = \sqrt{36} = 6$

3) $\sqrt{10} * \sqrt{12,1} = \sqrt{10 * 12,1} = \sqrt{121} = 11$

4) $\sqrt{0,5} * \sqrt{50} = \sqrt{0,5 * 50} = \sqrt{25} = 5$

5) $\sqrt{1\frac{3}{7}} * \sqrt{2,8} = \sqrt{1\frac{3}{7} * 2,8} = \sqrt{\frac{10}{7} * \frac{28}{10}} = \sqrt{\frac{1}{1} * \frac{4}{1}} = \sqrt{4} = 2$

6) $\sqrt{5 * 2^3} * \sqrt{5^3 * 2^3} = \sqrt{5 * 2^3 * 5^3 * 2^3} = \sqrt{5^{1 + 3} * 2^{3 + 3}} = \sqrt{5^{4} * 2^{6}} = \sqrt{5^4} * \sqrt{2^{6}} = \sqrt{(5^2)^2} * \sqrt{(2^3)^2} = |5^2| * |2^3| = |25| * |8| = 25 * 8 = 200$

502. Найдите значение выражения:
1) $\frac{\sqrt{75}}{\sqrt{3}}$;
2) $\frac{\sqrt{98}}{\sqrt{2}}$;
3) $\frac{\sqrt{3}}{\sqrt{48}}$;
4) $\frac{\sqrt{3,2}}{\sqrt{0,2}}$;
5) $\frac{\sqrt{72}}{\sqrt{50}}$;
6) $\frac{\sqrt{27}}{\sqrt{147}}$;
7) $\frac{\sqrt{6} * \sqrt{3}}{\sqrt{2}}$;
8) $\frac{\sqrt{5}}{\sqrt{3} * \sqrt{15}}$.

Решение:

1) $\frac{\sqrt{75}}{\sqrt{3}} = \sqrt{\frac{75}{3}} = \sqrt{25} = 5$

2) $\frac{\sqrt{98}}{\sqrt{2}} = \sqrt{\frac{98}{2}} = \sqrt{49} = 7$

3) $\frac{\sqrt{3}}{\sqrt{48}} = \sqrt{\frac{3}{48}} = \sqrt{\frac{1}{16}} = \frac{1}{4}$

4) $\frac{\sqrt{3,2}}{\sqrt{0,2}} = \sqrt{\frac{3,2}{0,2}} = \sqrt{\frac{32}{2}} = \sqrt{16} = 4$

5) $\frac{\sqrt{72}}{\sqrt{50}} = \sqrt{\frac{72}{50}} = \sqrt{\frac{36}{25}} = \frac{6}{5} = 1\frac{1}{5}$

6) $\frac{\sqrt{27}}{\sqrt{147}} = \sqrt{\frac{27}{147}} = \sqrt{\frac{9}{49}} = \frac{3}{7}$

7) $\frac{\sqrt{6} * \sqrt{3}}{\sqrt{2}} = \frac{\sqrt{6 * 3}}{\sqrt{2}} = \frac{\sqrt{18}}{\sqrt{2}} = \sqrt{\frac{18}{2}} = \sqrt{9} = 3$

8) $\frac{\sqrt{5}}{\sqrt{3} * \sqrt{15}} = \frac{\sqrt{5}}{\sqrt{3 * 15}} = \frac{\sqrt{5}}{\sqrt{45}} = \sqrt{\frac{5}{45}} = \sqrt{\frac{1}{9}} = \frac{1}{3}$