Степень с рациональным показателем 1. Вычислите: 1) 6-3.65 2) 45:48 3) (52)-1 4) 4() 5) 150 6) V8 7) √81 8) 9) V81.16 10) 11) √2-√8 12) √75 13) 25 14) -2.100; 15) 64 2. Упростите выражение: 1) 5-8.510-7-3:7-5+ () 5) yy 2) 4-27-0,3/16 + VI 6) 3) (6,7-10-3)-(5-10-2) 7) 4) 3. Представьте выражение •х в виде степени с основанием х. 4. Сократите дробь: 1) ; 2) . 5. Упростите выражение: 1) (a−b) - √b. 2) √√10-2-√√10+2 3) √24 4) 6. Решите уравнение. 35x-3= 27

1. Вычислите:

1. 6^-3 ⋅ 6⁵ = 6^-3+5 = 6² = 36
2. 4⁵ : 4⁸ = 4^5-8 = 4^-3 = \(\frac{1}{4^3}\) = \(\frac{1}{64}\)
3. (5²)^-1 = 5^2⋅(-1) = 5^-2 = \(\frac{1}{5^2}\) = \(\frac{1}{25}\)
4. 4 ⋅ \((\frac{1}{4}\))^-2 = 4 ⋅ 4² = 4 ⋅ 16 = 64
5. 15⁰ = 1
6. \(\sqrt{8}\) = \(\sqrt{2^3}\) = \(2^{\frac{3}{2}}\) = \(2^{1+\frac{1}{2}}\) = 2 ⋅ \(\sqrt{2}\)
7. \(\sqrt{81}\) = \(\sqrt{9^2}\) = 9
8. \(\sqrt[4]{\frac{1}{16}}\) = \(\sqrt[4]{\frac{1}{2^4}}\) = \(\frac{1}{2}\)
9. \(\sqrt[4]{81 ⋅ 16}\) = \(\sqrt[4]{81}\) ⋅ \(\sqrt[4]{16}\) = 3 ⋅ 2 = 6
10. \(\frac{\sqrt[3]{24}}{\sqrt[3]{3}}\) = \(\sqrt[3]{\frac{24}{3}}\) = \(\sqrt[3]{8}\) = 2
11. \(\sqrt{2} ⋅ \sqrt{8}\) = \(\sqrt{2 ⋅ 8}\) = \(\sqrt{16}\) = 4
12. \(\sqrt[5]{7^5}\) = 7
13. 25^{\(\frac{1}{2}\)} = \(\sqrt{25}\) = 5
14. -2 ⋅ 100^{\(\frac{1}{2}\)} = -2 ⋅ \(\sqrt{100}\) = -2 ⋅ 10 = -20
15. 64^{\(\frac{1}{3}\)} = \(\sqrt[3]{64}\) = 4

2. Упростите выражение:

1. 5^-8 ⋅ 5¹⁰ - 7^-3 : 7^-5 + \((\frac{3}{4}\))^-2 = 5² - 7² + \((\frac{4}{3}\))² = 25 - 49 + \(\frac{16}{9}\) = -24 + \(\frac{16}{9}\) = \(\frac{-216 + 16}{9}\) = \(\frac{-200}{9}\)
2. 4\(\sqrt[4]{-27}\) - 0.3\(\sqrt{16}\) + \(\sqrt[8]{1}\) = 4\(\sqrt[4]{-27}\) - 0.3 ⋅ 4 + 1 = 4\(\sqrt[4]{-27}\) - 1.2 + 1 = 4\(\sqrt[4]{-27}\) - 0.2
3. (6.7 ⋅ 10^-3) ⋅ (5 ⋅ 10^-2) = 6.7 ⋅ 5 ⋅ 10^-3 ⋅ 10^-2 = 33.5 ⋅ 10^-5 = 3.35 ⋅ 10^-4
4. \(\frac{7^{-7} ⋅ 7^{-8}}{7^{-13}}\) = \(\frac{7^{-15}}{7^{-13}}\) = 7^{-15 - (-13)} = 7^{-2} = \(\frac{1}{7^2}\) = \(\frac{1}{49}\)
5. y^{\(\frac{2}{7}\)} ⋅ y^{\(\frac{1}{6}\)} = y^{\(\frac{2}{7} + \frac{1}{6}\)} = y^{\(\frac{12+7}{42}\)} = y^{\(\frac{19}{42}\)}
6. \(\frac{x^{\frac{4}{9}} ⋅ x^{\frac{3}{5}}}{x^{\frac{5}{9}}}\) = \(\frac{x^{\frac{4}{9} + \frac{3}{5}}}{x^{\frac{5}{9}}}\) = \(\frac{x^{\frac{20+27}{45}}}{x^{\frac{5}{9}}}\) = \(\frac{x^{\frac{47}{45}}}{x^{\frac{25}{45}}}\) = x^{\(\frac{47}{45} - \frac{25}{45}\)} = x^{\(\frac{22}{45}\)}
7. (a⁸)^{\(\frac{7}{4}\)} ⋅ a^{\(\frac{7}{2}\)} = a^{8 ⋅ \(\frac{7}{4}\)} ⋅ a^{\(\frac{7}{2}\)} = a¹⁴ ⋅ a^{\(\frac{7}{2}\)} = a^{14 + \(\frac{7}{2}\)} = a^{\(\frac{28+7}{2}\)} = a^{\(\frac{35}{2}\)}

3. Представьте выражение в виде степени с основанием x:

4. Сократите дробь:

1. \(\frac{a^3 + b^3}{a^3 - b^3}\) = \(\frac{(a+b)(a^2 - ab + b^2)}{(a-b)(a^2 + ab + b^2)}\)
2. \(\frac{6c^2 - c}{c^2 - 6}\) = \(\frac{c(6c - 1)}{c^2 - 6}\)

5. Упростите выражение:

1. ((\(\frac{1}{a^4}\)) - b)² - \(\sqrt{b}\) = \(\frac{1}{a^8}\) - \(\frac{2b}{a^4}\) + b² - \(\sqrt{b}\)
2. \(\frac{\sqrt{(2 - 2\sqrt{2})^2} + \sqrt{(3 - 2\sqrt{2})^2}}{\sqrt{3} + \sqrt{5} ⋅ \sqrt{5-3}}\) = \(\frac{|2 - 2\sqrt{2}| + |3 - 2\sqrt{2}|}{\sqrt{3} + \sqrt{5} ⋅ \sqrt{2}}\) = \(\frac{2\sqrt{2} - 2 + 3 - 2\sqrt{2}}{\sqrt{3} + \sqrt{10}}\) = \(\frac{1}{\sqrt{3} + \sqrt{10}}\) = \(\frac{\sqrt{10} - \sqrt{3}}{7}\)
3. \(\frac{\sqrt{\sqrt{10} - 2} - \sqrt{\sqrt{10} + 2}}{\sqrt{24}}\) = \(\frac{\sqrt{\sqrt{10} - 2} - \sqrt{\sqrt{10} + 2}}{2\sqrt{6}}\) = \(\frac{\sqrt{6}}{12} ⋅ (\sqrt{\sqrt{10} + 2} - \sqrt{\sqrt{10} - 2})\)
4. \(\frac{(2a^2)^3 ⋅ (3b)^2}{(6a^3b)^2}\) = \(\frac{8a^6 ⋅ 9b^2}{36a^6b^2}\) = \(\frac{72a^6b^2}{36a^6b^2}\) = 2

6. Решите уравнение: