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10. 若$x^{3n}=2$，$n$是正整数，则$(3x^{3n})^3 + (-2x^{2n})^3$的值为（
A.248
B.184
C.150
D.98
B
）A.248
B.184
C.150
D.98
答案
10. B $(3x^{3n})^3 + (-2x^{2n})^3 = 27(x^{3n})^3 - 8x^{6n} = 27(x^{3n})^3 - 8(x^{3n})^2 = 27 × 2^3 - 8 × 2^2 = 216 - 32 = 184$.
11. 已知$ m $为奇数，$ n $为偶数，则下列各式计算正确的是（
A.$ (-3)^2 · (-3)^m = 3^{m+2} $
B.$ (-2)^3 · (-2)^m = -2^{m+3} $
C.$ (-4)^4 · (-4)^n = -4^{n+4} $
D.$ (-5)^5 · (-5)^n = -5^{n+5} $
D
）A.$ (-3)^2 · (-3)^m = 3^{m+2} $
B.$ (-2)^3 · (-2)^m = -2^{m+3} $
C.$ (-4)^4 · (-4)^n = -4^{n+4} $
D.$ (-5)^5 · (-5)^n = -5^{n+5} $
答案
11. D $(-3)^2 · (-3)^m = (-3)^{m+2}$,$\because m$为奇数,$\therefore m+2$为奇数,$\therefore (-3)^{m+2} = -3^{m+2}$,故A选项错误;$(-2)^3 · (-2)^m = (-2)^{m+3}$,$\because m$ 为奇数, $\therefore m + 3$ 为偶数,$\therefore (-2)^{m+3} = 2^{m+3}$,故 B 选项错误;$(-4)^4 · (-4)^n = (-4)^{n+4}$,$\because n$为偶数,$\therefore n + 4$为偶数,$\therefore (-4)^{n+4} = 4^{n+4}$,故C选项错误;$(-5)^5 · (-5)^n = (-5)^{n+5}$,$\because n$为偶数,$\therefore n + 5$为奇数,$\therefore (-5)^{n+5} = -5^{n+5}$,故D选项正确.
12. 现规定一种新运算“※”：$a※b = b^a$. 例如，$3※2 = 2^3 = 8$，则$3※(-\frac{1}{2}) =$
$-\frac{1}{8}$
，$4※(-2a^3b^2) =$ $16a^{12}b^8$
.答案
12. $-\frac{1}{8}$ $16a^{12}b^8$ $3※(-\frac{1}{2}) = (-\frac{1}{2})^3 = -\frac{1}{8}$,$4※(-2a^3b^2) = (-2a^3b^2)^4 = 16a^{12}b^8$.
13. 木星可以近似地看成一个球体，木星的半径约为 $7 × 10^4$ km，它的体积是
$1.436 × 10^{15}$
$ \mathrm{km}^3 $。（$ V_{\mathrm{球体}} = \frac{4}{3}π r^3 $，$π$ 取3.14，结果用科学记数法表示，结果精确到0.001）答案
13. $1.436 × 10^{15}$ 根据题意,得$V = \frac{4}{3}πr^3 \approx \frac{4}{3} × 3.14 × (7 × 10^4)^3 = \frac{4}{3} × 3.14 × 343 × 10^{12} \approx 1.436 × 10^{15} (\mathrm{km}^3)$.
14. (1)若$3 × 2^{x+3} × 4^{x+3} = 96$,求$x$的值;
(2)若$25^m × 2 × 10^n = 5^7 × 2^4$,求$mn$的值.
(2)若$25^m × 2 × 10^n = 5^7 × 2^4$,求$mn$的值.
答案
14. 解:(1)$\because 3 × 2^{x+3} × 4^{x+3} = 96$,
$\therefore 2^{x+3} × 2^{2(x+3)} = 32$,$\therefore 2^{3(x+3)} = 2^5$,
$\therefore 3(x + 3) = 5$,解得$x = -\frac{4}{3}$.
(2)$\because 25^m × 2 × 10^n = 5^7 × 2^4$,
$\therefore (5^2)^m × 2 × (2 × 5)^n = 5^7 × 2^4$,
$\therefore 5^{2m} × 2 × 2^n × 5^n = 5^7 × 2^4$,$\therefore 5^{2m+n} × 2^{n+1} = 5^7 × 2^4$,
$\therefore 2m + n = 7$,$n + 1 = 4$,解得$m = 2$,$n = 3$,$\therefore mn = 6$.
$\therefore 2^{x+3} × 2^{2(x+3)} = 32$,$\therefore 2^{3(x+3)} = 2^5$,
$\therefore 3(x + 3) = 5$,解得$x = -\frac{4}{3}$.
(2)$\because 25^m × 2 × 10^n = 5^7 × 2^4$,
$\therefore (5^2)^m × 2 × (2 × 5)^n = 5^7 × 2^4$,
$\therefore 5^{2m} × 2 × 2^n × 5^n = 5^7 × 2^4$,$\therefore 5^{2m+n} × 2^{n+1} = 5^7 × 2^4$,
$\therefore 2m + n = 7$,$n + 1 = 4$,解得$m = 2$,$n = 3$,$\therefore mn = 6$.
15. 新趋势·代数推理 试说明：$5^2 × 3^{2n+1} × 2^n - 3^n × 6^{n+2}$能被13整除.（n为自然数）
答案
15. 解:$\because 5^2 × 3^{2n+1} × 2^n - 3^n × 6^{n+2} = 5^2 × (3^{2n} × 3) × 2^n - 3^n × (6^n × 6^2) = 75 × 3^{2n} × 2^n - 36 × 3^n × 6^n = 75 × 9^n × 2^n - 36 × 3^n × 6^n = 75 × 18^n - 36 × 18^n = 39 × 18^n = 13 × 3 × 18^n$.
$\because 3 × 18^n$是整数,
$\therefore 5^2 × 3^{2n+1} × 2^n - 3^n × 6^{n+2}$能被13整除.
$\because 3 × 18^n$是整数,
$\therefore 5^2 × 3^{2n+1} × 2^n - 3^n × 6^{n+2}$能被13整除.
16. 推理能力 解决下列问题：
(1)若$a=4^5$，$b=5^4$，请用含$a$，$b$的代数式表示$20^{20}$；
(2)若$2^a=5^b=10$，请猜想$a+b$和$ab$的大小，并说明理由。
(1)若$a=4^5$，$b=5^4$，请用含$a$，$b$的代数式表示$20^{20}$；
(2)若$2^a=5^b=10$，请猜想$a+b$和$ab$的大小，并说明理由。
答案
16. 解:(1)$20^{20} = (4 × 5)^{20} = 4^{20} × 5^{20} = (4^5)^4 × (5^4)^5$.
将$a = 4^5$,$b = 5^4$代入,得$20^{20} = a^4 b^5$.
(2)$a + b = ab$. 理由如下:
$\because 2^a = 5^b = 10$,$\therefore (2^a)^b = 10^b$,$(5^b)^a = 10^a$,
$\therefore 2^{ab} = 10^b$,$5^{ab} = 10^a$,
$\therefore 2^{ab} · 5^{ab} = (2 × 5)^{ab} = 10^{ab} = 10^b · 10^a = 10^{a+b}$,
$\therefore a + b = ab$.
将$a = 4^5$,$b = 5^4$代入,得$20^{20} = a^4 b^5$.
(2)$a + b = ab$. 理由如下:
$\because 2^a = 5^b = 10$,$\therefore (2^a)^b = 10^b$,$(5^b)^a = 10^a$,
$\therefore 2^{ab} = 10^b$,$5^{ab} = 10^a$,
$\therefore 2^{ab} · 5^{ab} = (2 × 5)^{ab} = 10^{ab} = 10^b · 10^a = 10^{a+b}$,
$\therefore a + b = ab$.
