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1. (2025·陕西)计算$2a^{2}· ab$的结果为(
A.$4a^{2}b$
B.$4a^{3}b$
C.$2a^{2}b$
D.$2a^{3}b$
D
)A.$4a^{2}b$
B.$4a^{3}b$
C.$2a^{2}b$
D.$2a^{3}b$
答案
1. D
解析
$2a^{2}·ab=2×1× a^{2+1}b=2a^{3}b$，结果为D。
2. 有下列算式：①$3a^{3}· (2a^{2})^{2}=12a^{12}$；②$(2× 10^{3})× (\frac{1}{2}× 10^{3})=10^{6}$；③$-3xy· (-2xyz)^{2}=12x^{3}y^{3}z^{2}$；④$4x^{3}· 5x^{4}=9x^{12}$.其中，计算正确的个数是(
A.0
B.1
C.2
D.3
B
)A.0
B.1
C.2
D.3
答案
2. B
解析
①$3a^{3}·(2a^{2})^{2}=3a^{3}·4a^{4}=12a^{7}≠12a^{12}$；
②$(2×10^{3})×(\frac{1}{2}×10^{3})=(2×\frac{1}{2})×(10^{3}×10^{3})=1×10^{6}=10^{6}$；
③$-3xy·(-2xyz)^{2}=-3xy·4x^{2}y^{2}z^{2}=-12x^{3}y^{3}z^{2}≠12x^{3}y^{3}z^{2}$；
④$4x^{3}·5x^{4}=20x^{7}≠9x^{12}$。
正确的只有②，个数是1。
B
②$(2×10^{3})×(\frac{1}{2}×10^{3})=(2×\frac{1}{2})×(10^{3}×10^{3})=1×10^{6}=10^{6}$；
③$-3xy·(-2xyz)^{2}=-3xy·4x^{2}y^{2}z^{2}=-12x^{3}y^{3}z^{2}≠12x^{3}y^{3}z^{2}$；
④$4x^{3}·5x^{4}=20x^{7}≠9x^{12}$。
正确的只有②，个数是1。
B
3. (1)(2025·秦淮区月考)填空：(
(2)(2025·秦淮区期中)已知单项式$3x^{2}y^{3}$与$2xy^{2}$的积为$mx^{3}y^{n}$，则$m-n=$
$9x^{2}y$
)$× 2xy=18x^{3}y^{2}$；(2)(2025·秦淮区期中)已知单项式$3x^{2}y^{3}$与$2xy^{2}$的积为$mx^{3}y^{n}$，则$m-n=$
1
.答案
3. (1) $9x^{2}y$ (2) 1
4. 某长方体的长为$4× 10^{7}\ \mathrm{cm}$，宽为$3× 10^{5}\ \mathrm{cm}$，高为$2× 10^{3}\ \mathrm{cm}$，则长方体的体积是
$2.4×10^{22}$
$\mathrm{cm}^{3}$.答案
4. $2.4×10^{22}$
解析
长方体体积 = 长×宽×高 = $(4×10^{7})×(3×10^{5})×(2×10^{3})$
$=(4×3×2)×(10^{7}×10^{5}×10^{3})$
$=24×10^{15}$
$=2.4×10^{16}\ \mathrm{cm}^{3}$
1
$=(4×3×2)×(10^{7}×10^{5}×10^{3})$
$=24×10^{15}$
$=2.4×10^{16}\ \mathrm{cm}^{3}$
1
5. 计算：
(1)$-\frac{2}{3}a^{2}b· \frac{5}{6}ac^{2}$；
(2)$16x^{9}y^{2}· (-x^{5}y^{15})$；
(3)$4xy^{2}· (-\frac{3}{8}x^{2}yz^{3})$；
(4)$3.2mn^{2}· (-0.125m^{2}n^{3})$.
(1)$-\frac{2}{3}a^{2}b· \frac{5}{6}ac^{2}$；
(2)$16x^{9}y^{2}· (-x^{5}y^{15})$；
(3)$4xy^{2}· (-\frac{3}{8}x^{2}yz^{3})$；
(4)$3.2mn^{2}· (-0.125m^{2}n^{3})$.
答案
5. (1) $-\frac{2}{3}a^{2}b· \frac{5}{6}ac^{2}=(-\frac{2}{3}×\frac{5}{6})·(a^{2}·a)·b·c^{2}=-\frac{5}{9}a^{3}bc^{2}$；
(2) $16x^{9}y^{2}· (-x^{5}y^{15})=16×(-1)·(x^{9}·x^{5})·(y^{2}·y^{15})=-16x^{14}y^{17}$；
(3) $4xy^{2}· (-\frac{3}{8}x^{2}yz^{3})=4×(-\frac{3}{8})·(x·x^{2})·(y^{2}·y)·z^{3}=-\frac{3}{2}x^{3}y^{3}z^{3}$；
(4) $3.2mn^{2}· (-0.125m^{2}n^{3})=3.2×(-0.125)·(m·m^{2})·(n^{2}·n^{3})=-0.4m^{3}n^{5}$。
(2) $16x^{9}y^{2}· (-x^{5}y^{15})=16×(-1)·(x^{9}·x^{5})·(y^{2}·y^{15})=-16x^{14}y^{17}$；
(3) $4xy^{2}· (-\frac{3}{8}x^{2}yz^{3})=4×(-\frac{3}{8})·(x·x^{2})·(y^{2}·y)·z^{3}=-\frac{3}{2}x^{3}y^{3}z^{3}$；
(4) $3.2mn^{2}· (-0.125m^{2}n^{3})=3.2×(-0.125)·(m·m^{2})·(n^{2}·n^{3})=-0.4m^{3}n^{5}$。
解析
(1) $-\frac{2}{3}a^{2}b· \frac{5}{6}ac^{2}=(-\frac{2}{3}×\frac{5}{6})·(a^{2}·a)·b·c^{2}=-\frac{5}{9}a^{3}bc^{2}$；
(2) $16x^{9}y^{2}· (-x^{5}y^{15})=16×(-1)·(x^{9}·x^{5})·(y^{2}·y^{15})=-16x^{14}y^{17}$；
(3) $4xy^{2}· (-\frac{3}{8}x^{2}yz^{3})=4×(-\frac{3}{8})·(x·x^{2})·(y^{2}·y)·z^{3}=-\frac{3}{2}x^{3}y^{3}z^{3}$；
(4) $3.2mn^{2}· (-0.125m^{2}n^{3})=3.2×(-0.125)·(m·m^{2})·(n^{2}·n^{3})=-0.4m^{3}n^{5}$。
(2) $16x^{9}y^{2}· (-x^{5}y^{15})=16×(-1)·(x^{9}·x^{5})·(y^{2}·y^{15})=-16x^{14}y^{17}$；
(3) $4xy^{2}· (-\frac{3}{8}x^{2}yz^{3})=4×(-\frac{3}{8})·(x·x^{2})·(y^{2}·y)·z^{3}=-\frac{3}{2}x^{3}y^{3}z^{3}$；
(4) $3.2mn^{2}· (-0.125m^{2}n^{3})=3.2×(-0.125)·(m·m^{2})·(n^{2}·n^{3})=-0.4m^{3}n^{5}$。
6. 计算：
(1)$(-\frac{1}{2}xyz)· \frac{2}{3}x^{2}y^{2}· (-\frac{3}{5}yz^{3})$；
(2)$5x· (-\frac{1}{3}ax)· (-2.25axy)· 1.2x^{2}y^{2}$；
(3)$(2x^{3}y)^{2}· (-2xy)$；
(4)$-6m^{2}n· (x-y)^{3}· \frac{1}{3}mn^{2}(y-x)^{2}$；
(5)$-(a^{2}b)^{3}+2a^{2}b· (-3a^{2}b)^{2}$；
(6)$(-2a^{2})^{3}+2a^{2}· a^{4}-a^{8}÷ a^{2}$.
(1)$(-\frac{1}{2}xyz)· \frac{2}{3}x^{2}y^{2}· (-\frac{3}{5}yz^{3})$；
(2)$5x· (-\frac{1}{3}ax)· (-2.25axy)· 1.2x^{2}y^{2}$；
(3)$(2x^{3}y)^{2}· (-2xy)$；
(4)$-6m^{2}n· (x-y)^{3}· \frac{1}{3}mn^{2}(y-x)^{2}$；
(5)$-(a^{2}b)^{3}+2a^{2}b· (-3a^{2}b)^{2}$；
(6)$(-2a^{2})^{3}+2a^{2}· a^{4}-a^{8}÷ a^{2}$.
答案
6. (1) $(-\frac{1}{2}xyz)· \frac{2}{3}x^{2}y^{2}· (-\frac{3}{5}yz^{3})$
$=(-\frac{1}{2}×\frac{2}{3}×(-\frac{3}{5}))·(x·x^{2})·(y·y^{2}·y)·(z·z^{3})$
$=\frac{1}{5}x^{3}y^{4}z^{4}$
(2) $5x· (-\frac{1}{3}ax)· (-2.25axy)· 1.2x^{2}y^{2}$
$=5×(-\frac{1}{3})×(-2.25)×1.2·a·a·x·x·x·x^{2}·y·y^{2}$
$=4.5a^{2}x^{5}y^{3}$
(3) $(2x^{3}y)^{2}· (-2xy)$
$=4x^{6}y^{2}· (-2xy)$
$=-8x^{7}y^{3}$
(4) $-6m^{2}n· (x-y)^{3}· \frac{1}{3}mn^{2}(y-x)^{2}$
$=-6×\frac{1}{3}·m^{2}·m·n·n^{2}·(x-y)^{3}·(x-y)^{2}$
$=-2m^{3}n^{3}(x - y)^{5}$
(5) $-(a^{2}b)^{3}+2a^{2}b· (-3a^{2}b)^{2}$
$=-a^{6}b^{3}+2a^{2}b·9a^{4}b^{2}$
$=-a^{6}b^{3}+18a^{6}b^{3}$
$=17a^{6}b^{3}$
(6) $(-2a^{2})^{3}+2a^{2}· a^{4}-a^{8}÷ a^{2}$
$=-8a^{6}+2a^{6}-a^{6}$
$=-7a^{6}$
$=(-\frac{1}{2}×\frac{2}{3}×(-\frac{3}{5}))·(x·x^{2})·(y·y^{2}·y)·(z·z^{3})$
$=\frac{1}{5}x^{3}y^{4}z^{4}$
(2) $5x· (-\frac{1}{3}ax)· (-2.25axy)· 1.2x^{2}y^{2}$
$=5×(-\frac{1}{3})×(-2.25)×1.2·a·a·x·x·x·x^{2}·y·y^{2}$
$=4.5a^{2}x^{5}y^{3}$
(3) $(2x^{3}y)^{2}· (-2xy)$
$=4x^{6}y^{2}· (-2xy)$
$=-8x^{7}y^{3}$
(4) $-6m^{2}n· (x-y)^{3}· \frac{1}{3}mn^{2}(y-x)^{2}$
$=-6×\frac{1}{3}·m^{2}·m·n·n^{2}·(x-y)^{3}·(x-y)^{2}$
$=-2m^{3}n^{3}(x - y)^{5}$
(5) $-(a^{2}b)^{3}+2a^{2}b· (-3a^{2}b)^{2}$
$=-a^{6}b^{3}+2a^{2}b·9a^{4}b^{2}$
$=-a^{6}b^{3}+18a^{6}b^{3}$
$=17a^{6}b^{3}$
(6) $(-2a^{2})^{3}+2a^{2}· a^{4}-a^{8}÷ a^{2}$
$=-8a^{6}+2a^{6}-a^{6}$
$=-7a^{6}$
解析
(1) $(-\frac{1}{2}xyz)· \frac{2}{3}x^{2}y^{2}· (-\frac{3}{5}yz^{3})$
$=(-\frac{1}{2}×\frac{2}{3}×(-\frac{3}{5}))·(x·x^{2})·(y·y^{2}·y)·(z·z^{3})$
$=\frac{1}{5}x^{3}y^{4}z^{4}$
(2) $5x· (-\frac{1}{3}ax)· (-2.25axy)· 1.2x^{2}y^{2}$
$=5×(-\frac{1}{3})×(-2.25)×1.2·a·a·x·x·x·x^{2}·y·y^{2}$
$=4.5a^{2}x^{5}y^{3}$
(3) $(2x^{3}y)^{2}· (-2xy)$
$=4x^{6}y^{2}· (-2xy)$
$=-8x^{7}y^{3}$
(4) $-6m^{2}n· (x-y)^{3}· \frac{1}{3}mn^{2}(y-x)^{2}$
$=-6×\frac{1}{3}·m^{2}·m·n·n^{2}·(x-y)^{3}·(x-y)^{2}$
$=-2m^{3}n^{3}(x - y)^{5}$
(5) $-(a^{2}b)^{3}+2a^{2}b· (-3a^{2}b)^{2}$
$=-a^{6}b^{3}+2a^{2}b·9a^{4}b^{2}$
$=-a^{6}b^{3}+18a^{6}b^{3}$
$=17a^{6}b^{3}$
(6) $(-2a^{2})^{3}+2a^{2}· a^{4}-a^{8}÷ a^{2}$
$=-8a^{6}+2a^{6}-a^{6}$
$=-7a^{6}$
$=(-\frac{1}{2}×\frac{2}{3}×(-\frac{3}{5}))·(x·x^{2})·(y·y^{2}·y)·(z·z^{3})$
$=\frac{1}{5}x^{3}y^{4}z^{4}$
(2) $5x· (-\frac{1}{3}ax)· (-2.25axy)· 1.2x^{2}y^{2}$
$=5×(-\frac{1}{3})×(-2.25)×1.2·a·a·x·x·x·x^{2}·y·y^{2}$
$=4.5a^{2}x^{5}y^{3}$
(3) $(2x^{3}y)^{2}· (-2xy)$
$=4x^{6}y^{2}· (-2xy)$
$=-8x^{7}y^{3}$
(4) $-6m^{2}n· (x-y)^{3}· \frac{1}{3}mn^{2}(y-x)^{2}$
$=-6×\frac{1}{3}·m^{2}·m·n·n^{2}·(x-y)^{3}·(x-y)^{2}$
$=-2m^{3}n^{3}(x - y)^{5}$
(5) $-(a^{2}b)^{3}+2a^{2}b· (-3a^{2}b)^{2}$
$=-a^{6}b^{3}+2a^{2}b·9a^{4}b^{2}$
$=-a^{6}b^{3}+18a^{6}b^{3}$
$=17a^{6}b^{3}$
(6) $(-2a^{2})^{3}+2a^{2}· a^{4}-a^{8}÷ a^{2}$
$=-8a^{6}+2a^{6}-a^{6}$
$=-7a^{6}$
