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8. 已知$3^{x+2} · 5^{x+2}=15^{3x-4}$，求$(x-1)^2 -3x(x-2)-4$的值.
答案
8.解: $\because 3^{x+2} · 5^{x+2}=15^{x+2}=15^{3x-4},$
$\therefore x+2=3x-4,解得 x=3.$
$\therefore (x-1)^2 -3x(x-2)-4$
$=(3-1)^2 -3×3×(3-2)-4$
$=4-9-4=-9.$
$\therefore x+2=3x-4,解得 x=3.$
$\therefore (x-1)^2 -3x(x-2)-4$
$=(3-1)^2 -3×3×(3-2)-4$
$=4-9-4=-9.$
【推理能力】规定:若两数$a,b$满足$a^m=b$,则记为$(a,b)=m$.例如:因为$2^3=8$,所以记为$(2,8)=3$.我们还可以利用该规定来说明等式$(3,3)+(3,9)=(3,27)$成立,理由如下:设$(3,3)=m$, $(3,9)=n$,则$3^m=3,3^n=9$,故$3^m×3^n=3^{m+n}=3×9=27$,则$(3,27)=m+n$,即$(3,3)+(3,9)=(3,27)$.
(1)根据上述规定填空:$(6,36)=\_\_\_\_\_\_$.
(2)如果$(3,m+17)=4,(4,m)=n$,那么$(5,\_\_\_\_\_\_)=n$.
(3)若$(3^n,2^n)=s,(3,2)=t$,请说明$s$与$t$的关系.($n$为正整数)
(1)根据上述规定填空:$(6,36)=\_\_\_\_\_\_$.
(2)如果$(3,m+17)=4,(4,m)=n$,那么$(5,\_\_\_\_\_\_)=n$.
(3)若$(3^n,2^n)=s,(3,2)=t$,请说明$s$与$t$的关系.($n$为正整数)
答案
(1)2 (2)125
点拨:根据材料,知由$(3,m+17)=4$,可得$3^4=m+17, \therefore m=64$,由$(4,m)=n$,可得$4^n=m=64$,即$n=3, \therefore 5^n=5^3=125.$
(3)解: $\because (3^n,2^n)=s, \therefore 3^{ns}=2^n. \because (3,2)=t,$
$\therefore 3^t=2, \therefore 3^{tn}=2^n, \therefore 3^{ns}=3^{tn}, \therefore s=t.$
点拨:根据材料,知由$(3,m+17)=4$,可得$3^4=m+17, \therefore m=64$,由$(4,m)=n$,可得$4^n=m=64$,即$n=3, \therefore 5^n=5^3=125.$
(3)解: $\because (3^n,2^n)=s, \therefore 3^{ns}=2^n. \because (3,2)=t,$
$\therefore 3^t=2, \therefore 3^{tn}=2^n, \therefore 3^{ns}=3^{tn}, \therefore s=t.$
