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一、选择题
1. 化简 $ x^{3} · ( \dfrac{y^{3}}{x} )^{2} $ 的结果是(
A.$ xy^{6} $
B.$ xy^{5} $
C.$ x^{2}y^{5} $
D.$ x^{2}y^{6} $
1. 化简 $ x^{3} · ( \dfrac{y^{3}}{x} )^{2} $ 的结果是(
A
)。A.$ xy^{6} $
B.$ xy^{5} $
C.$ x^{2}y^{5} $
D.$ x^{2}y^{6} $
答案
1. A
解析
$x^{3} · ( \dfrac{y^{3}}{x} )^{2} = x^{3} · \dfrac{y^{6}}{x^{2}} = x y^{6}$，答案选A。
2. 计算 $ ( -\dfrac{a^{3}}{2b} )^{2} ÷ ( -\dfrac{a^{2}}{b} )^{3} × ( \dfrac{b}{2} )^{2} $ 的结果是(
A.$ -\dfrac{1}{16} $
B.$ 16b^{2} $
C.$ -\dfrac{b^{3}}{16} $
D.$ \dfrac{b^{3}}{16} $
C
)。A.$ -\dfrac{1}{16} $
B.$ 16b^{2} $
C.$ -\dfrac{b^{3}}{16} $
D.$ \dfrac{b^{3}}{16} $
答案
2. C
解析
$\begin{aligned}(-\dfrac{a^{3}}{2b})^{2} ÷ (-\dfrac{a^{2}}{b})^{3} × (\dfrac{b}{2})^{2}&=\dfrac{a^{6}}{4b^{2}} ÷ (-\dfrac{a^{6}}{b^{3}}) × \dfrac{b^{2}}{4}\\&=\dfrac{a^{6}}{4b^{2}} × (-\dfrac{b^{3}}{a^{6}}) × \dfrac{b^{2}}{4}\\&=-\dfrac{b}{4} × \dfrac{b^{2}}{4}\\&=-\dfrac{b^{3}}{16}\end{aligned}$
C
C
3. 下列计算正确的是(
A.$ \dfrac{(a + b)^{2}}{2} = \dfrac{a^{2} + b^{2}}{4} $
B.$ ( \dfrac{-2x^{2}}{a - b} )^{2} = \dfrac{4x^{2}}{(a - b)^{2}} $
C.$ ( \dfrac{x^{2} - 2y^{3}}{-5} )^{3} = \dfrac{x^{6} - 8y^{3}}{125} $
D.$ ( \dfrac{yz^{2}}{2x^{n}} )^{3} = \dfrac{y^{3}z^{6}}{8x^{3n}} $
D
)。A.$ \dfrac{(a + b)^{2}}{2} = \dfrac{a^{2} + b^{2}}{4} $
B.$ ( \dfrac{-2x^{2}}{a - b} )^{2} = \dfrac{4x^{2}}{(a - b)^{2}} $
C.$ ( \dfrac{x^{2} - 2y^{3}}{-5} )^{3} = \dfrac{x^{6} - 8y^{3}}{125} $
D.$ ( \dfrac{yz^{2}}{2x^{n}} )^{3} = \dfrac{y^{3}z^{6}}{8x^{3n}} $
答案
3. D
二、填空题
4. $ \dfrac{9x}{2y} · ( -\dfrac{2y}{3x} )^{3} = $
4. $ \dfrac{9x}{2y} · ( -\dfrac{2y}{3x} )^{3} = $
$-\frac{4 y^{2}}{3 x^{2}}$
。答案
4. $-\frac{4 y^{2}}{3 x^{2}}$
5. $ ( -\dfrac{x^{2}}{y} )^{5} · ( \dfrac{y^{2}}{-x} )^{2} · ( \dfrac{1}{xy} )^{7} = $
$-\frac{x}{y^{8}}$
。答案
5. $-\frac{x}{y^{8}}$
解析
$(-\dfrac{x^{2}}{y})^{5}·(\dfrac{y^{2}}{-x})^{2}·(\dfrac{1}{xy})^{7}$
$=(-1)^5·\dfrac{(x^{2})^{5}}{y^{5}}·\dfrac{(y^{2})^{2}}{(-x)^{2}}·\dfrac{1^{7}}{(xy)^{7}}$
$=-\dfrac{x^{10}}{y^{5}}·\dfrac{y^{4}}{x^{2}}·\dfrac{1}{x^{7}y^{7}}$
$=-\dfrac{x^{10}·y^{4}·1}{y^{5}·x^{2}·x^{7}y^{7}}$
$=-\dfrac{x^{10}}{x^{9}y^{8}}$
$=-\dfrac{x}{y^{8}}$
$=(-1)^5·\dfrac{(x^{2})^{5}}{y^{5}}·\dfrac{(y^{2})^{2}}{(-x)^{2}}·\dfrac{1^{7}}{(xy)^{7}}$
$=-\dfrac{x^{10}}{y^{5}}·\dfrac{y^{4}}{x^{2}}·\dfrac{1}{x^{7}y^{7}}$
$=-\dfrac{x^{10}·y^{4}·1}{y^{5}·x^{2}·x^{7}y^{7}}$
$=-\dfrac{x^{10}}{x^{9}y^{8}}$
$=-\dfrac{x}{y^{8}}$
6. $ ( \dfrac{-x}{x - y} )^{2} · \dfrac{x^{2} - y^{2}}{x} = $
$\frac{x^{2}+x y}{x-y}$
。答案
6. $\frac{x^{2}+x y}{x-y}$
解析
$(\dfrac{-x}{x - y})^{2} · \dfrac{x^{2} - y^{2}}{x}$
$=\dfrac{x^{2}}{(x - y)^{2}} · \dfrac{(x + y)(x - y)}{x}$
$=\dfrac{x(x + y)}{x - y}$
$=\dfrac{x^{2} + xy}{x - y}$
$=\dfrac{x^{2}}{(x - y)^{2}} · \dfrac{(x + y)(x - y)}{x}$
$=\dfrac{x(x + y)}{x - y}$
$=\dfrac{x^{2} + xy}{x - y}$
三、解答题
7. 化简：$ -( -\dfrac{a^{2}}{b} )^{2} · ( -\dfrac{b^{3}}{a} )^{3} ÷ (ab)^{4} $。
7. 化简：$ -( -\dfrac{a^{2}}{b} )^{2} · ( -\dfrac{b^{3}}{a} )^{3} ÷ (ab)^{4} $。
答案
7. $\frac{b^{3}}{a^{3}}$
解析
$ - ( -\dfrac{a^{2}}{b} )^{2} · ( -\dfrac{b^{3}}{a} )^{3} ÷ (ab)^{4} $
$ = - ( \dfrac{a^{4}}{b^{2}} ) · ( -\dfrac{b^{9}}{a^{3}} ) ÷ (a^{4}b^{4}) $
$ = - \dfrac{a^{4}}{b^{2}} · ( -\dfrac{b^{9}}{a^{3}} ) · \dfrac{1}{a^{4}b^{4}} $
$ = \dfrac{a^{4}b^{9}}{b^{2}a^{3}} · \dfrac{1}{a^{4}b^{4}} $
$ = \dfrac{ab^{7}}{a^{4}b^{4}} $
$ = \dfrac{b^{3}}{a^{3}} $
$ = - ( \dfrac{a^{4}}{b^{2}} ) · ( -\dfrac{b^{9}}{a^{3}} ) ÷ (a^{4}b^{4}) $
$ = - \dfrac{a^{4}}{b^{2}} · ( -\dfrac{b^{9}}{a^{3}} ) · \dfrac{1}{a^{4}b^{4}} $
$ = \dfrac{a^{4}b^{9}}{b^{2}a^{3}} · \dfrac{1}{a^{4}b^{4}} $
$ = \dfrac{ab^{7}}{a^{4}b^{4}} $
$ = \dfrac{b^{3}}{a^{3}} $
8. 化简：$ (-2a^{3}b^{3}) · \dfrac{-b^{2}}{(-2a)^{2}} ÷ (a^{3}b^{2})^{2} $。
答案
8. $\frac{b}{2 a^{5}}$
解析
$(-2a^{3}b^{3}) · \dfrac{-b^{2}}{(-2a)^{2}} ÷ (a^{3}b^{2})^{2}$
$=(-2a^{3}b^{3}) · \dfrac{-b^{2}}{4a^{2}} ÷ (a^{6}b^{4})$
$=(-2a^{3}b^{3}) · \dfrac{-b^{2}}{4a^{2}} · \dfrac{1}{a^{6}b^{4}}$
$=\dfrac{(-2)×(-1)}{4} · a^{3-2-6} · b^{3+2-4}$
$=\dfrac{2}{4} · a^{-5} · b^{1}$
$=\dfrac{1}{2}a^{-5}b$
$=\dfrac{b}{2a^{5}}$
$=(-2a^{3}b^{3}) · \dfrac{-b^{2}}{4a^{2}} ÷ (a^{6}b^{4})$
$=(-2a^{3}b^{3}) · \dfrac{-b^{2}}{4a^{2}} · \dfrac{1}{a^{6}b^{4}}$
$=\dfrac{(-2)×(-1)}{4} · a^{3-2-6} · b^{3+2-4}$
$=\dfrac{2}{4} · a^{-5} · b^{1}$
$=\dfrac{1}{2}a^{-5}b$
$=\dfrac{b}{2a^{5}}$
9. 先化简，再求值：$ \dfrac{a^{2} - ab}{a + b} · ( \dfrac{a + b}{a - b} )^{2} ÷ \dfrac{a}{a - b} $，其中 $ a + b = 100 $。
答案
9. 100
解析
$\begin{aligned}&\frac{a^{2} - ab}{a + b} · ( \frac{a + b}{a - b} )^{2} ÷ \frac{a}{a - b}\\=&\frac{a(a - b)}{a + b} · \frac{(a + b)^2}{(a - b)^2} · \frac{a - b}{a}\\=&\frac{a(a - b)(a + b)^2(a - b)}{(a + b)(a - b)^2a}\\=&a + b\\\end{aligned}$
当$a + b = 100$时，原式$=100$
100
当$a + b = 100$时，原式$=100$
100
