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
