Question

Numărul natural x care verifică egalitatea: a) (3^x)^5=3^10×3^5
b) x+(2^2)^5=2^3×2^7
c) 5^3×6+5^4×7
d) x^2+x^3+x^4
e) 3^x+3^x-1=4
f) 5×3^x-2×3^×=81
g) 3^2×+1-3^2×-3^2×-1-3^2×-2-3^2×-3=123
Ajutatima va rog mult

Answer 1

Răspuns:

Explicație pas cu pas:

a)  

( 3ˣ)⁵ = 3¹⁰ x 3⁵

3⁵ˣ = 3⁽¹⁰⁺⁵⁾

3⁵ˣ = 3¹⁵ =>    5 x = 15 =>  x = 15 : 5 =>   x = 3

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b)

x + ( 2²)⁵ =2³ x 2⁷

x + 2²ˣ⁵ = 2³⁺⁷

x + 2¹⁰ = 2¹⁰

x = 2¹⁰ - 2¹⁰

x = 0

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c)

5³ x 6 + 5⁴ x 7 = 5³ x ( 6 + 5 x 7 ) = 5³ x 41 = 125 x 41 = 5 125

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d)

x² + x³ + x⁴ = x² ( 1 + x + x²)

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e)

3ˣ + 3ˣ⁻¹ = 4

3ˣ( 1 + 3⁻1) = 4

3ˣ ( 3/3 + 1/3) = 4

3ˣ x (4/3 ) = 4

3ˣ = 4 : (4/3)

3ˣ = 4 x ( 3/4)

3ˣ = 3¹ =>    x = 1

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f)

5 x 3ˣ - 2 x 3ˣ = 81

3ˣ x ( 5 - 2 ) = 81

3ˣ x 3 = 81

3ˣ = 81 : 3

3ˣ = 27

3ˣ = 3³ =>   x = 3

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g)  

3²ˣ⁺¹ - 3²ˣ - 3²ˣ⁻¹ - 3²ˣ⁻² - 3²ˣ⁻³ = 123

3²ˣ x ( 3¹ - 1 - 3⁻¹ - 3⁻² - 3⁻³ ) = 123

3²ˣ x ( 2 - 1/3 - 1/3² - 1/3³ ) = 123

3²ˣ x ( 54/27 - 9/27 - 3/27 - 1/27 ) = 123

3²ˣ x ( 41/27 ) = 123

3²ˣ = 123 : ( 41/27 )

3²ˣ = 123 x ( 27/41)

3²ˣ = 3 x 3³

3²ˣ = 3¹⁺³

3²ˣ = 3⁴ =>   2 x = 4 =>    x = 4 : 2 =>   x = 2