551 Вправа 551Алгебра 8 клас Тарасенкова 2025 · § 13 · стор. 161

551. Порівняйте числа:

1) ${\displaystyle 4\sqrt{3}=}$${\displaystyle \sqrt{4^2\cdot 3}=}$${\displaystyle \sqrt{16\cdot 3}=}$${\displaystyle \sqrt{48}}$; ${\displaystyle 7=}$${\displaystyle \sqrt{7^2}=}$${\displaystyle \sqrt{49}}$;

${\displaystyle \sqrt{48}<\sqrt{49}}$, отже, ${\displaystyle 4\sqrt{3}<7}$;

2) ${\displaystyle 2\sqrt{6}=}$${\displaystyle \sqrt{2^2\cdot 6}=}$${\displaystyle \sqrt{4\cdot 6}=}$${\displaystyle \sqrt{24}}$;

${\displaystyle \sqrt{28}>\sqrt{24}}$, отже, ${\displaystyle \sqrt{28}>2\sqrt{6}}$;

3) ${\displaystyle 3\sqrt{5}=}$${\displaystyle \sqrt{3^2\cdot 5}=}$${\displaystyle \sqrt{9\cdot 5}=}$${\displaystyle \sqrt{45}}$; ${\displaystyle 7=}$${\displaystyle \sqrt{7^2}=}$${\displaystyle \sqrt{49}}$;

${\displaystyle \sqrt{45}<\sqrt{49}}$, отже, ${\displaystyle 3\sqrt{5}<7}$;

4) ${\displaystyle 3\sqrt{2}=}$${\displaystyle \sqrt{3^2\cdot 2}=}$${\displaystyle \sqrt{9\cdot 2}=}$${\displaystyle \sqrt{18}}$;

${\displaystyle \sqrt{18}=}$${\displaystyle \sqrt{18}}$, отже, ${\displaystyle 3\sqrt{2}=}$${\displaystyle \sqrt{18}}$;

5) ${\displaystyle 5\sqrt{2}=}$${\displaystyle \sqrt{5^2\cdot 2}=}$${\displaystyle \sqrt{25\cdot 2}=}$${\displaystyle \sqrt{50}}$; ${\displaystyle 7=}$${\displaystyle \sqrt{7^2}=}$${\displaystyle \sqrt{49}}$;

${\displaystyle \sqrt{50}>\sqrt{49}}$, отже, ${\displaystyle 5\sqrt{2}>7}$;

6) ${\displaystyle 2\sqrt{3}=}$${\displaystyle \sqrt{2^2\cdot 3}=}$${\displaystyle \sqrt{4\cdot 3}=}$${\displaystyle \sqrt{12}}$; ${\displaystyle 3\sqrt{2}=}$${\displaystyle \sqrt{3^2\cdot 2}=}$${\displaystyle \sqrt{9\cdot 2}=}$${\displaystyle \sqrt{18}}$;

${\displaystyle \sqrt{12}<\sqrt{18}}$, отже, ${\displaystyle 2\sqrt{3}<3\sqrt{2}}$;

7) ${\displaystyle 5\sqrt{6}=}$${\displaystyle \sqrt{5^2\cdot 6}=}$${\displaystyle \sqrt{25\cdot 6}=}$${\displaystyle \sqrt{150}}$; ${\displaystyle 6\sqrt{5}=}$${\displaystyle \sqrt{6^2\cdot 5}=}$${\displaystyle \sqrt{36\cdot 5}=}$${\displaystyle \sqrt{180}}$;

${\displaystyle \sqrt{150}<\sqrt{180}}$, отже, ${\displaystyle 5\sqrt{6}<6\sqrt{5}}$;

8) ${\displaystyle 14\sqrt{2}=}$${\displaystyle \sqrt{{14}^2\cdot 2}=}$${\displaystyle \sqrt{196\cdot 2}=}$${\displaystyle \sqrt{392}}$; ${\displaystyle 8\sqrt{7}=}$${\displaystyle \sqrt{8^2\cdot 7}=}$${\displaystyle \sqrt{64\cdot 7}=}$${\displaystyle \sqrt{448}}$;

${\displaystyle \sqrt{392}<\sqrt{448}}$, отже, ${\displaystyle 14\sqrt{2}<8\sqrt{7}}$;

9) ${\displaystyle 2\sqrt{0,01}=}$${\displaystyle \sqrt{2^2\cdot 0,01}=}$${\displaystyle \sqrt{4\cdot 0,01}=}$${\displaystyle \sqrt{0,04}}$;

${\displaystyle \sqrt{0,04}<\sqrt{0,09}}$, отже, ${\displaystyle 2\sqrt{0,01}<\sqrt{0,09}}$;

10) ${\displaystyle 0,25=}$${\displaystyle \frac{1}{4}=}$${\displaystyle \frac{9}{36}}$; ${\displaystyle \frac{4}{9}=}$${\displaystyle \frac{16}{36}}$;

${\displaystyle \sqrt{\frac{4}{9}}>\sqrt{0,25}}$, отже, ${\displaystyle \sqrt{\frac{4}{9}}>\sqrt{0,25}}$;

11) ${\displaystyle 1\frac{9}{16}=}$${\displaystyle \frac{25}{16}<2}$;

${\displaystyle \sqrt{1\frac{9}{16}}<\sqrt{2}}$, отже, ${\displaystyle \sqrt{1\frac{9}{16}}<\sqrt{2}}$;

12) ${\displaystyle \frac{1}{0,01}=}$${\displaystyle 100}$;

${\displaystyle \sqrt{100}<\sqrt{105}}$, отже, ${\displaystyle \sqrt{\frac{1}{0,01}}<\sqrt{105}}$.