Răspuns:
- (B) n = 1006
- (C) n = 55
- (D) n = 10
Explicație pas cu pas:
Salutare!
(B)
[tex]\bf 9^{n}+9^{n+1}= 10\cdot 3^{2012}[/tex]
[tex]\bf 9^{n}\cdot( 9^{n-n}+9^{n+1-n})= 10\cdot 3^{2012}[/tex]
[tex]\bf (3^{2})^{n}\cdot( 9^{0}+9^{1})= 10\cdot 3^{2012}[/tex]
[tex]\bf 3^{2\cdot n}\cdot 10= 10\cdot 3^{2012}\:\:\:\Big|:10[/tex]
[tex]\bf 3^{2n} = 3^{2012}\implies2n=2012\:\:\Big|:2\implies\boxed{\bf n = 1006}[/tex]
(C)
[tex]\bf 6^{n}+6^{n+3}= 217\cdot 6^{55}[/tex]
[tex]\bf 6^{n}\cdot( 6^{n-n}+6^{n+3-n})= 217\cdot 6^{55}[/tex]
[tex]\bf 6^{n}\cdot( 6^{0}+6^{3})= 217\cdot 6^{55}[/tex]
[tex]\bf 6^{n}\cdot( 1+2016)= 217\cdot 6^{55}[/tex]
[tex]\bf 6^{n}\cdot 217= 217\cdot 6^{55}\:\:\:\Big|:217[/tex]
[tex]\bf 6^{n}=6^{55}\implies \boxed{\bf n = 55}[/tex]
(D)
[tex]\bf 7^{n+1}+7^{n+2}= 8\cdot 7^{11}[/tex]
[tex]\bf 7^{n+1}\cdot(7^{n+1-(n+1)}+7^{n+2-(n+1)})= 8\cdot 7^{11}[/tex]
[tex]\bf 7^{n+1}\cdot(7^{0}+7^{1})= 8\cdot 7^{11}[/tex]
[tex]\bf 7^{n+1}\cdot(1+7)= 8\cdot 7^{11}[/tex]
[tex]\bf 7^{n+1}\cdot 8 = 8\cdot 7^{11}\:\:\:\Big|:8[/tex]
[tex]\bf 7^{n+1} = 7^{11} \implies n+1 = 11 \implies n=11-1 \implies \boxed{\bf n = 10}[/tex]
Formule pentru puteri
a⁰ = 1 sau 1 = a⁰
(aⁿ)ᵇ = aⁿ ˣ ᵇ sau aⁿ ˣ ᵇ = (aⁿ) ᵇ
aⁿ · aᵇ = (a · a) ⁿ ⁺ ᵇ sau (a · a) ⁿ ⁺ ᵇ = aⁿ · aᵇ
aⁿ : aᵇ = (a : a) ⁿ ⁻ ᵇ sau (a : a) ⁿ ⁻ ᵇ = aⁿ : aᵇ
aⁿ · bⁿ = (a · b)ⁿ sau (a · b)ⁿ = aⁿ · bⁿ
aⁿ : bⁿ = (a : b)ⁿ sau (a : b)ⁿ = aⁿ : bⁿ
#copaceibrainly
