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8. 下列各式中，计算结果不等于$2^{11}$的是(
A.$2^{10}+2^{10}$
B.$2^{12}-2^{10}$
C.$2^{7}× 2^{4}$
D.$2^{15}÷ 2^{4}$
B
)A.$2^{10}+2^{10}$
B.$2^{12}-2^{10}$
C.$2^{7}× 2^{4}$
D.$2^{15}÷ 2^{4}$
答案
8. B 解析：$2^{10}+2^{10}=2×2^{10}=2^{11}$，故A选项不符合题意；$2^{12}$与$2^{10}$不是同类项，不能合并，故B选项符合题意；$2^{7}×2^{4}=2^{7 + 4}=2^{11}$，故C选项不符合题意；$2^{15}÷2^{4}=2^{15 - 4}=2^{11}$，故D选项不符合题意.
9. 若$m$为正整数，计算$3^{2m}· 4^{2m}· 36^{m}÷ 72^{2m}$的结果为(
A.$1$
B.$-1$
C.$2$
D.$3$
A
)A.$1$
B.$-1$
C.$2$
D.$3$
答案
9. A 解析：原式$=12^{2m}·6^{2m}÷72^{2m}=(12×6÷72)^{2m}=1$.
10. 若$4^{x}=a$，$8^{y}=b$，则$2^{2x - 3y}$可表示为
$\frac{a}{b}$
.(用含$a$、$b$的代数式表示)答案
10. $\frac{a}{b}$ 解析：$2^{2x - 3y}=2^{2x}÷2^{3y}=4^{x}÷8^{y}=\frac{a}{b}$.
解析
$2^{2x - 3y}=2^{2x}÷2^{3y}=(2^{2})^{x}÷(2^{3})^{y}=4^{x}÷8^{y}=\frac{a}{b}$
11. (1)若$32÷ 8^{n - 1}=2^{n}$，则$n=$
(2)若$2^{x}÷ 16^{y}=8$，则$2x - 8y=$
2
.(2)若$2^{x}÷ 16^{y}=8$，则$2x - 8y=$
6
.答案
11. (1)2 解析：$\because32÷8^{n - 1}=2^{n}$，$\therefore(2^{5})÷(2^{3})^{n - 1}=2^{n}$，$\therefore2^{5}÷2^{3n - 3}=2^{n}$，$\therefore2^{5 - (3n - 3)}=2^{n}$，$5-(3n - 3)=n$，解得$n = 2$. (2)6 解析：由题意，得$2^{x}÷2^{4y}=2^{3}$，$\therefore x - 4y = 3$，$\therefore$原式$=2(x - 4y)=2×3 = 6$.
解析
(1) $\because 32÷ 8^{n - 1}=2^{n}$
$\therefore 2^{5}÷ (2^{3})^{n - 1}=2^{n}$
$\therefore 2^{5}÷ 2^{3n - 3}=2^{n}$
$\therefore 2^{5 - (3n - 3)}=2^{n}$
$\therefore 5 - (3n - 3)=n$
$5 - 3n + 3 = n$
$8 = 4n$
$n = 2$
(2) $\because 2^{x}÷ 16^{y}=8$
$\therefore 2^{x}÷ (2^{4})^{y}=2^{3}$
$\therefore 2^{x}÷ 2^{4y}=2^{3}$
$\therefore 2^{x - 4y}=2^{3}$
$\therefore x - 4y = 3$
$\therefore 2x - 8y = 2(x - 4y)=2× 3 = 6$
$\therefore 2^{5}÷ (2^{3})^{n - 1}=2^{n}$
$\therefore 2^{5}÷ 2^{3n - 3}=2^{n}$
$\therefore 2^{5 - (3n - 3)}=2^{n}$
$\therefore 5 - (3n - 3)=n$
$5 - 3n + 3 = n$
$8 = 4n$
$n = 2$
(2) $\because 2^{x}÷ 16^{y}=8$
$\therefore 2^{x}÷ (2^{4})^{y}=2^{3}$
$\therefore 2^{x}÷ 2^{4y}=2^{3}$
$\therefore 2^{x - 4y}=2^{3}$
$\therefore x - 4y = 3$
$\therefore 2x - 8y = 2(x - 4y)=2× 3 = 6$
12. 计算：
(1)$(x - 2y)^{4}÷ (2y - x)^{2}÷ (x - 2y)$；
(2)$(x + y)^{10}÷ (-x - y)^{6}÷ (x + y)^{3}$.
(1)$(x - 2y)^{4}÷ (2y - x)^{2}÷ (x - 2y)$；
(2)$(x + y)^{10}÷ (-x - y)^{6}÷ (x + y)^{3}$.
答案
12. (1)原式$=(x - 2y)^{4}÷(x - 2y)^{2}÷(x - 2y)=(x - 2y)^{4 - 2 - 1}=x - 2y$. (2)原式$=(x + y)^{10}÷(x + y)^{6}÷(x + y)^{3}=(x + y)^{10 - 6 - 3}=x + y$.
13. (1)已知$2^{m}=3$，$2^{n}=5$，求$2^{3m}÷ 2^{2n}$的值.
(2)已知$10^{α }=20$，$10^{β }=\frac{1}{5}$，求$25^{α }÷ 5^{2β }$的值.
(2)已知$10^{α }=20$，$10^{β }=\frac{1}{5}$，求$25^{α }÷ 5^{2β }$的值.
答案
13. (1)$\because2^{m}=3$，$2^{n}=5$，$\therefore2^{3m}÷2^{2n}=(2^{m})^{3}÷(2^{n})^{2}=3^{3}÷5^{2}=\frac{27}{25}$. (2)$\because10^{α}=20$，$10^{β}=\frac{1}{5}$，$\therefore10^{α}÷10^{β}=10^{α - β}=20÷\frac{1}{5}=100=10^{2}$，$\thereforeα - β = 2$，$\therefore25^{α}÷25^{2β}=25^{α}÷25^{2β}=25^{α - β}=25^{2}=625$.
14. 已知$4^{m + 3}· 8^{m + 1}÷ 2^{4m + 7}=16$，求$m$的值.
答案
14. $\because4^{m + 3}·8^{m + 1}÷2^{4m + 7}=16$，$\therefore(2^{2})^{m + 3}·(2^{3})^{m + 1}÷2^{4m + 7}=2^{4}$，$\therefore2^{2(m + 3)}·2^{3(m + 1)}÷2^{4m + 7}=2^{4}$，$\therefore2^{2(m + 3)+3(m + 1)-(4m + 7)}=2^{4}$，$\therefore2^{m + 2}=2^{4}$，$\therefore m + 2 = 4$，解得$m = 2$.
15. 已知$4^{m}÷ 2^{n}=8$，$(2^{m})^{2}· 2^{n}=32$.
(1)求$2m - n$的值.
(2)计算：$(-8)^{2m + n}× 0.125^{2m - n}$.
(1)求$2m - n$的值.
(2)计算：$(-8)^{2m + n}× 0.125^{2m - n}$.
答案
15. (1)$\because4^{m}÷2^{n}=8$，$\therefore2^{2m}÷2^{n}=2^{3}$，$\therefore2^{2m - n}=2^{3}$，$\therefore2m - n = 3$. (2)$\because(2^{m})^{2}·2^{n}=32$，$\therefore2^{2m}·2^{n}=2^{5}$，$\therefore2^{2m + n}=2^{5}$，$\therefore2m + n = 5$，$\therefore(-8)^{2m + n}×0.125^{2m - n}=(-8)^{5}×(\frac{1}{8})^{3}=(-8)^{2}×[(-8)×\frac{1}{8}]^{3}=64×(-1)=-64$.
16. 已知$2^{a}=3$，$2^{b}=5$，$2^{c}=75$.
(1)求$2^{c + b - a}$的值.
(2)试说明：$a = c - 2b$.
(1)求$2^{c + b - a}$的值.
(2)试说明：$a = c - 2b$.
答案
16. (1)$2^{c + b - a}=2^{c}·2^{b}÷2^{a}=75×5÷3=125$. (2)$\because2^{a}=3$，$2^{c - 2b}=2^{c}÷2^{2b}=2^{c}÷(2^{b})^{2}=75÷5^{2}=3$，$\therefore2^{a}=2^{c - 2b}$，$\therefore a = c - 2b$.
