Математика,помогите с примерами

I.

1. (21³ ⋅ 40⁴) / (24⁴ ⋅ 35³) = ((3⋅7)³ ⋅ (2³⋅5)⁴) / ((2³⋅3)⁴ ⋅ (5⋅7)³) = (3³⋅7³⋅2¹²⋅5⁴) / (2¹²⋅3⁴⋅5³⋅7³) = 3³⁻⁴ ⋅ 7³⁻³ ⋅ 2¹²⁻¹² ⋅ 5⁴⁻³ = 3⁻¹ ⋅ 5¹ = 5/3

2. (a³ ⋅ (a⋅b)¹⁰) / (a⁹ ⋅ (b⁵)² ⋅ a⁴) = (a³ ⋅ a¹⁰ ⋅ b¹⁰) / (a⁹ ⋅ b¹⁰ ⋅ a⁴) = a¹³⋅b¹⁰ / a¹³⋅b¹⁰ = 1

II.

1. ∛(3⁶) ⋅ ∛64 = 3⁶ᐟ³ ⋅ ∛(4³) = 3² ⋅ 4 = 9 ⋅ 4 = 36

2. ⁸√(64 ⋅ 14² ⋅ 7⁶) = ⁸√(2⁶ ⋅ (2⋅7)² ⋅ 7⁶) = ⁸√(2⁶ ⋅ 2² ⋅ 7² ⋅ 7⁶) = ⁸√(2⁸ ⋅ 7⁸) = ⁸√(14⁸) = 14

3. (1/3)⁻³ ⋅ 9⁻² = (3⁻¹)⁻³ ⋅ (3²)⁻² = 3³ ⋅ 3⁻⁴ = 3⁻¹ = 1/3

4. 16⁻³ᐟ⁴ = (2⁴)⁻³ᐟ⁴ = 2⁻³ = 1/8

5. (∛250 ⋅ √4) / (√25 ⋅ ∛16) = (∛(125⋅2) ⋅ 2) / (5 ⋅ ∛(8⋅2)) = (5∛2 ⋅ 2) / (5 ⋅ 2∛2) = 1

6. (3⁶⋅6⁵)/(6⁴⋅3³) − (7⁸⋅3³)/(3⋅7⁷) = 3⁶⁻³⋅6⁵⁻⁴ − 7⁸⁻⁷⋅3³⁻¹ = 3³⋅6¹ − 7¹⋅3² = 27⋅6 − 7⋅9 = 162 − 63 = 99

7. 4¹ᐟ⁶ ⋅ 4⁵ᐟ⁶ = 4¹ᐟ⁶⁺⁵ᐟ⁶ = 4⁶ᐟ⁶ = 4¹ = 4

III.

1. 9/⁷√(3⁵) = (9/⁷√(3⁵)) ⋅ (⁷√(3²)/⁷√(3²)) = 9⁷√(3²)/⁷√(3⁷) = 9⁷√9/3 = 3⁷√9

2. 4/(√3+√7) = 4/(√7+√3) ⋅ (√7−√3)/(√7−√3) = 4(√7−√3)/((√7)²−(√3)²) = 4(√7−√3)/(7−3) = 4(√7−√3)/4 = √7−√3

IV.

1. √(3x² + 11x − 26) = x + 2

3x² + 11x − 26 = (x + 2)²

3x² + 11x − 26 = x² + 4x + 4

2x² + 7x − 30 = 0

D = 7² − 4⋅2⋅(−30) = 49 + 240 = 289

x = (−7 ± √289)/4 = (−7 ± 17)/4

x₁ = 10/4 = 2.5

x₂ = −24/4 = −6

Условие: x + 2 ≥ 0 ⇒ x ≥ −2

x₁ = 2.5: 2.5 ≥ −2 ✓

x₂ = −6: −6 < −2 ✗

x = 2.5

2. √(−8 + 3x) = x − 4

−8 + 3x = (x − 4)²

−8 + 3x = x² − 8x + 16

x² − 11x + 24 = 0

(x − 3)(x − 8) = 0

x₁ = 3, x₂ = 8

Условие: x − 4 ≥ 0 ⇒ x ≥ 4

x₁ = 3: 3 < 4 ✗

x₂ = 8: 8 ≥ 4 ✓

x = 8

3. ∛(x² − 17) = 4

x² − 17 = 4³

x² − 17 = 64

x² = 81

x = ±9