Question

Determinați numărul natural x care verifica egalitatile

Answer 1

Răspuns:

Explicație pas cu pas:

17

a) 3⁽ˣ⁺²⁾ - 5 x 3ˣ = 36

3ˣ x 3² - 5 x 3ˣ = 36

3ˣ ( 3² - 5 ) = 36

3ˣ( 9 - 5) = 36

3ˣ x 4 = 36

3ˣ = 36 : 4

3ˣ = 9

3ˣ = 3² ⇒    x = 2

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b) 2⁵¹ + 2⁵² + 2⁵³ = x × 2⁵⁰

2⁵¹× ( 1 + 2¹ + 2²) = x × 2⁵⁰  l  : 2⁵⁰

2 × 7 = x

x = 14

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c)  5⁽ˣ⁺²⁾ × 2⁽ˣ⁺¹⁾ + 7×10ˣ = 57 000

5ˣ × 5² × 2ˣ × 2¹ + 7 × 10ˣ = 57 000

(5×2)ˣ × 25 × 2 + 7 × 10ˣ = 57 000

10ˣ × 50 + 7 × 10ˣ = 57 000

10ˣ × ( 50 + 7 ) = 57 000

10ˣ × 57 = 57 000 l : 57

10ˣ = 1 000

10ˣ = 10³  ⇒    x = 3

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d) 12 × 4ˣ - 5 × 4ˣ = 7 × 2¹²

4ˣ × ( 12 - 5 ) = 7 × 2¹²

4ˣ × 7 = 7 × 2¹²   l  : 7

2²ˣ = 2¹²

2x = 12 ⇒   x = 12 : 2 ⇒   x = 6

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e)  4⁽ˣ⁺²⁾ + 4⁽ˣ⁺¹⁾ + 4ˣ = 336

4ˣ × 4² + 4ˣ × 4¹ + 4ˣ = 336

4ˣ × ( 4² + 4 + 1 ) = 336

4ˣ × ( 16+5) = 336

4ˣ × 21 = 336

4ˣ = 336 : 21

4ˣ = 16

4ˣ = 4² ⇒   x = 2

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f) 9⁴³ - 9⁴² - 9⁴¹ = 71 × 3ˣ

9⁴¹ × ( 9² - 9¹ - 1 ) = 71 × 3ˣ

9⁴¹ × ( 81 - 9 - 1 ) = 71 × 3ˣ

9⁴¹ × 71 = 71 × 3ˣ  l : 71

(3²)⁴¹ = 3ˣ

3²ˣ⁴¹  = 3ˣ ⇒    x = 2×41 ⇒   x = 82

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g) 7⁽ˣ⁺³⁾ - 7⁽ˣ⁺²⁾ - 7⁽ˣ⁺¹⁾ = 41 × 7²ˣ

7ˣ × 7³ - 7ˣ × 7² - 7ˣ × 7¹ = 41 × 7²ˣ

7ˣ × ( 7³ - 7² - 7¹) = 41 × 7²ˣ

7ˣ × 7 × ( 7² - 7 - 1 ) = 41 × 7²ˣ

7⁽ˣ ⁺¹⁾ × 41 = 41 × 7²ˣ   l  : 41

7⁽ˣ⁺¹⁾ = 7²ˣ ⇒  2x = x + 1 ⇒  2x - x = 1 ⇒  x = 1

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h) 3⁽ˣ⁺³⁾ × 4⁽ˣ⁺¹⁾ - 5×3ˣ ×4ˣ = 1 236

3ˣ × 3³× 4ˣ × 4¹ - 5 × (3×4)ˣ = 1236

(3×4)ˣ × 27×4 - 12ˣ × 5 = 1 236

12ˣ × ( 108 - 5 ) = 1 236

12ˣ × 103 = 1 236  

12ˣ = 1 236 : 103

12ˣ = 12  ⇒   x = 1