1. Вычислите: 1) √0,25; 2) √32; 3) ∛-3 ⅜; 4) 0,7∜81; 5) ∜¹⁶/₈₁ + ∛-¹/₈; 6) (2∜4)⁶; 7) ⁶/(₂√₃)²; 8) -3∛(-7)³; 9) √16 ⋅√-125; 10) √₁⋅∛0,008; 11) √2⋅∜8; 12) ⁵√54⋅⁵√32; 13) ∛¹⁸⁹/₃∛⁷ ; 14) (2¹²⁄₅ ⋅2⁸⁄₅)^½ 2. Найти значение выражения: 2^(√2+1) ÷ 2²√²; ((√6)^√²)√²; ∛²√²⋅√8;

1. Вычислите:

1) √0,25

√0,25 = 0,5

2) √32

√32 = √(16*2) = √16 * √2 = 4√2

3) ∛-3 ⅜

∛-3 ⅜ = ∛(-27/8) = -3/2 = -1,5

4) 0,7∜81

0,7 * ∜81 = 0,7 * 3 = 2,1

5) ∜¹⁶/₈₁ + ∛-¹/₈

∜¹⁶/₈₁ + ∛-¹/₈ = 2/3 - 1/2 = 4/6 - 3/6 = 1/6

6) (2∜4)⁶

(2∜4)⁶ = 2⁶ * (4^(1/4))⁶ = 64 * 4^(6/4) = 64 * 4^(3/2) = 64 * (√4)³ = 64 * 2³ = 64 * 8 = 512

7) ⁶/(₂√₃)²

⁶/(₂√₃)² = 6 / (4 * 3) = 6 / 12 = 1/2

8) -3∛(-7)³

-3∛(-7)³ = -3 * (-7) = 21

9) √16 ⋅√-125

√16 ⋅√-125 = 4 * √(-1 * 125) = 4 * i * √125 = 4 * i * √(25 * 5) = 4 * i * 5√5 = 20i√5, где i - мнимая единица

10) √₁⋅∛0,008

√₁⋅∛0,008 = 1 * 0,2 = 0,2

11) √2⋅∜8

√2⋅∜8 = 2^(1/2) * 8^(1/4) = 2^(1/2) * (2³) ^(1/4) = 2^(1/2) * 2^(3/4) = 2^(1/2 + 3/4) = 2^(5/4) = 2^(1 + 1/4) = 2∜2

12) ⁵√54⋅⁵√32

⁵√54⋅⁵√32 = ⁵√(54 * 32) = ⁵√(2 * 27 * 32) = ⁵√(2 * 27 * 2 * 16) = ⁵√(4 * 27 * 16) = ⁵√(4 * 27 * 16) = ⁵√(1728)

13) ∛¹⁸⁹/₃∛⁷

∛¹⁸⁹/₃∛⁷ = ∛(189 / 7) / 3 = ∛27 / 3 = 3 / 3 = 1

14) (2¹²⁄₅ ⋅2⁸⁄₅)^½

(2¹²⁄₅ ⋅2⁸⁄₅)^½ = (2^(12/5 + 8/5))^(1/2) = (2^(20/5))^(1/2) = (2⁴)^(1/2) = 16^(1/2) = √16 = 4

2. Найти значение выражения:

2^(√2+1) ÷ 2²√²; ((√6)^√²)√²; ∛²√²⋅√8

2^(√2+1) ÷ 2²√² = 2^(√2 + 1 - 2√2) = 2^(1 - √2)

((√6)^√²)√² = (√6)^(√2 * √2) = (√6)² = 6

∛²√²⋅√8 = (2√2)^(1/3) * (2³)^(1/2) = 2^(√2/3) * 2^(3/2) = 2^(√2/3 + 3/2) = 2^((2√2 + 9)/6)