1. [2025青岛模拟] 下列计算正确的是 ()

A.$a + 2a = 3a^2$
B.$a^5 ÷ a^2 = a^3$
C.$(-a)^2 · a^3 = -a^5$
D.$(2a^3)^2 = 2a^6$

$a + 2a = 3a$,A 选项错误;$(-a)^{2}· a^{3}=a^{2}· a^{3}=a^{5}$,C选项错误;$(2a^{3})^{2}=4a^{6}$,D 选项错误.

变式 计算：
(1) $a^7 · 2a + (a^2)^4 - (-3a^4)^2$；
(2) $-(-a^2)^3 · (-a)^3 ÷ (-a^2)$；
(3) $(-a^2)^2 · a^5 + a^{10} ÷ a - (-2a^3)^3$。

(1) $a^7 · 2a + (a^2)^4 - (-3a^4)^2 = a^7 · 2a + a^8 -9a^8 = 2a^8 + a^8 -9a^8 = -6a^8$.
(2) $-(-a^2)^3 · (-a)^3 ÷ (-a^2) = -(-a^6) · (-a^3) ÷ (-a^2) = a^6 · (-a^3) ÷ (-a^2) = -a^9 ÷ (-a^2) = a^7$.
(3) $(-a^2)^2 · a^5 + a^{10} ÷ a - (-2a^3)^3 = a^4 · a^5 + a^9 - (-8a^9) = a^9 + a^9 + 8a^9 = 10a^9$.

2. 若$2^a = 3, 2^b = 6, 2^c = 12$，请说明：
(1)$b = a + 1$；
(2)$c = a + 2$；
(3)$a + c = 2b$；
(4)$b + c = 2a + 3$

(1)$\because 2^b ÷ 2^a = 2^{b-a} =6 ÷3 =2$,
$\therefore b -a =1,\therefore b = a + 1$.
(2)$\because 2^c ÷ 2^a = 2^{c-a} =12 ÷3 =4=2^2$,
$\therefore c -a =2,\therefore c = a + 2$.
(3)$\because 2^a ×2^c =2^{a+c}=3×12=36=(2^b)^2=2^{2b}$,
$\therefore a + c = 2b$.
(4)$\because 2^b ×2^c =2^{b+c}=6×12=(2^a)^2 ×2^3=2^{2a+3}$,
$\therefore b + c = 2a + 3$.

3. 若$a=2^{10},b=3^{8},c=5^{6}$,请比较$a,b,c$的大小.

$\because a=2^{10},b=3^{8},c=5^{6}$,
$\therefore a=(2^5)^2=32^2,b=(3^4)^2=81^2,c=5^6=(5^3)^2=125^2$(化成指数相同的幂,比较底数的大小).
$\because 32 ＜81 ＜125,\therefore a ＜ b ＜ c$.

变式 请比较$3^{57}$和$6^{34}$的大小.

$\because 3^{57}=3^{5×11}×3^2=(3^5)^{11}×9=243^{11}×9,6^{34}=6^{3×11}×6=(6^3)^{11}×6=216^{11}×6,243＞216,9＞6$,
$\therefore 3^{57}＞6^{34}$.