$\\$

`a,`

`(33^7 +33^7 + 33^7 +33^7)/(7^5 + 7^5 + 7^5 + 7^5 + 7^5)`

`= (4. 33^7)/(5 . 7^5)`

`= 4/5 . 33^7/7^5`

`(21^5 + 21^5 + 21^5 + 21^5 +21^5)/(11^7 + 11^7 + 11^7 + 11^7)`

`= (5 . 21^5)/(4 . 11^7)`

`= 5/4 . 21^5/11^7`

`(33^7 +33^7 + 33^7 +33^7)/(7^5 + 7^5 + 7^5 + 7^5 + 7^5) . (21^5 + 21^5 + 21^5 + 21^5 +21^5)/(11^7 + 11^7 + 11^7 + 11^7) =3^x`

`-> 4/5 . 33^7/7^5 . 5/4 . 21^5/11^7 = 3^x`

`-> (4/5 . 5/4) . 33^7/7^5 . 21^5/11^7 = 3^x`

`-> (11^7 . 3^7)/7^5 . (3^5 . 7^5)/11^7 =3^x`

`-> 3^7/3^5 = 3^x`

`-> 3^2=3^x`

`-> x=2` (Thỏa mãn)

Vậy `x=2`

$\\$
`b,`

`13^{n+2} +11^{n+1} - 13^{n+1} +11^n`

`= 13^n . 13^2 +11^n . 11 - 13^n . 13 +11^n`

`= (13^n . 13^2 - 13^n . 13) + (11^n . 11 +11^n)`

`= 13^n . (13^2 - 13) + 11^n . (11+1)`

`= 13^n . 156 +11^n . 12`

`= 13^{n+1} . 156 : 13 + 11^n . 12`

`= 13^{n+1} . 12+11^n . 12`

`= 12 . (13^{n+1}+11^n)`

Vì `12 \vdots 12`

Do `n` là số nguyên dương

`->12 . (13^{n+1} +11^n) \vdots 12` với mọi `n` nguyên dương

`-> 13^{n+2} +11^{n+1} - 13^{n+1} +11^n \vdots 12` với mọi `n` nguyên dương

$\\$

`c,`

`7^{(x+1) (x-3)}=1`

`-> 7^{(x+1)(x-3)}=7^0`

`->(x+1)(x-3)=0`

`->` \(\left[ \begin{array}{l}x+1=0\\x-3=0\end{array} \right.\) $\\$ `->` \(\left[ \begin{array}{l}x=-1\\x=3\end{array} \right.\) 

Vậy `x=-1` hoặc `x=3`

$\\$

`d,`

`3^x +3^{x+1}+3^{x+2}+3^{x+3}=360`

`->3^x +3^x . 3 + 3^x . 3^2 +3^x . 3^3=360`

`->3^x . (1+3+3^2+3^3) = 360`

`->3^x . (1+3+9+27)=360`

`->3^x . 40=360`

`->3^x=9`

`->3^x=3^2`

`->x=2`

Vậy `x=2`

Đáp án+Giải thích các bước giải:

`a`

\(\dfrac{33^7 +33^7 +33^7 +33^7}{7^5 +7^5 +7^5 +7^5 +7^5}\times \dfrac{21^5 +21^5 +21^5 +21^5 +21^5}{11^7+11^7+11^7+11^7}=3^x\\\Leftrightarrow \dfrac{4\times 33^7}{5\times 7^5}\times \dfrac{5\times 21^5}{4\times 11^7}=3^x\\\Leftrightarrow \dfrac{3^7 \times 11^7}{7^5}\times \dfrac{3^5 \times 7^5}{11^7}=3^x\\\Leftrightarrow 3^7 \times 3^5=3^x\\\Leftrightarrow 3^{12}=3^x\\\Leftrightarrow x=12\)

`b`

\(13^{n+2}+11^{n+1}-13^{n+1}+11^n\\= (13-1)13^{n+1} + (11+1)11^n\\= 12\times 13^{n+1}+ 12\times 11^n\\=12(13^{n+1}\times 11^n)\vdots12\)

`c`

\(7^{(x+1)(x-3)}=1\\\Leftrightarrow 7^{(x+1)(x-3)}=7^0\\\Leftrightarrow (x+1)(x-3)=0\\\Leftrightarrow \left[ \begin{array}{l}x+1=0\\x-3=0\end{array} \right.\\\Leftrightarrow \left[ \begin{array}{l}x=-1\\x=3\end{array} \right.\)

`d`

\(3^x + 3^{x+1}+3^{x+2}+3^{x+3}=360\\\Leftrightarrow (1+3+3^2+3^3)3^x =360\\\Leftrightarrow 40\times 3^x =360\\\Leftrightarrow 3^x =9\\\Leftrightarrow 3^x= 3^2\\\Leftrightarrow x=2\)