8.4E：练习

练习成就完美

练习集 A：使用以下方法简化表达式\(a^{\frac{1}{n}}\)

在以下练习中，写成激进表达式。

1. a.\(x^{\frac{1}{2}}\) b.\(y^{\frac{1}{3}}\) c.\(z^{\frac{1}{4}}\)
2. a.\(r^{\frac{1}{2}}\) b.\(s^{\frac{1}{3}}\) c.\(t^{\frac{1}{4}}\)
3. a.\(u^{\frac{1}{5}}\) b.\(v^{\frac{1}{9}}\) c.\(w^{\frac{1}{20}}\)
4. a.\(g^{\frac{1}{7}}\) b.\(h^{\frac{1}{5}}\) c.\(j^{\frac{1}{25}}\)

回答

1. a.\(\sqrt{x}\) b.\(\sqrt[3]{y}\) c.\(\sqrt[4]{z}\)

3. a.\(\sqrt[5]{u}\) b.\(\sqrt[9]{v}\) c.\(\sqrt[20]{w}\)

练习集 B：使用以下方法简化表达式\(a^{\frac{1}{n}}\)

在以下练习中，用有理指数书写。

1. a.\(\sqrt[7]{x}\) b.\(\sqrt[9]{y}\) c.\(\sqrt[5]{f}\)
2. a.\(\sqrt[8]{4}\) b.\(\sqrt[10]{s}\) c.\(\sqrt[4]{t}\)
3. a.\(\sqrt[3]{7c}\) b.\(\sqrt[7]{12d}\) c.\(2\sqrt[4]{6b}\)
4. a.\(\sqrt[4]{5x}\) b.\(\sqrt[8]{9y}\) c.\(7\sqrt[5]{3z}\)
5. a.\(\sqrt{21p}\) b.\(\sqrt[4]{8q}\) c.\(4\sqrt[6]{36r}\)
6. a.\(\sqrt[3]{25a}\) b.\(\sqrt{3b}\) c.\(\sqrt[8]{40c}\)

回答

1. a.\(x^{\frac{1}{7}}\) b.\(y^{\frac{1}{9}}\) c.\(f^{\frac{1}{5}}\)

3. a.\((7 c)^{\frac{1}{4}}\) b.\((12 d)^{\frac{1}{7}}\) c.\(2(6 b)^{\frac{1}{4}}\)

5. a.\((21 p)^{\frac{1}{2}}\) b.\((8 q)^{\frac{1}{4}}\) c.\(4(36 r)^{\frac{1}{6}}\)

练习 SET C：使用以下方法简化表达式\(a^{\frac{1}{n}}\)

在以下练习中，进行简化。

1. a.\(81^{\frac{1}{2}}\) b.\(125^{\frac{1}{3}}\) c.\(64^{\frac{1}{2}}\)
2. a.\(625^{\frac{1}{4}}\) b.\(243^{\frac{1}{5}}\) c.\(32^{\frac{1}{5}}\)
3. a.\(16^{\frac{1}{4}}\) b.\(16^{\frac{1}{2}}\) c.\(625^{\frac{1}{4}}\)
4. a.\(64^{\frac{1}{3}}\) b.\(32^{\frac{1}{5}}\) c.\(81^{\frac{1}{4}}\)
5. a.\((-216)^{\frac{1}{3}}\) b.\(-216^{\frac{1}{3}}\) c.\((216)^{-\frac{1}{3}}\)
6. a.\((-1000)^{\frac{1}{3}}\) b.\(-1000^{\frac{1}{3}}\) c.\((1000)^{-\frac{1}{3}}\)
7. a.\((-81)^{\frac{1}{4}}\) b.\(-81^{\frac{1}{4}}\) c.\((81)^{-\frac{1}{4}}\)
8. a.\((-49)^{\frac{1}{2}}\) b.\(-49^{\frac{1}{2}}\) c.\((49)^{-\frac{1}{2}}\)
9. a.\((-36)^{\frac{1}{2}}\) b.\(-36^{\frac{1}{2}}\) c.\((36)^{-\frac{1}{2}}\)
10. a.\((-16)^{\frac{1}{4}}\) b.\(-16^{\frac{1}{4}}\) c.\(16^{-\frac{1}{4}}\)
11. a.\((-100)^{\frac{1}{2}}\) b.\(-100^{\frac{1}{2}}\) c.\((100)^{-\frac{1}{2}}\)
12. a.\((-32)^{\frac{1}{5}}\) b.\((243)^{-\frac{1}{5}}\) c.\(-125^{\frac{1}{3}}\)

回答

1. a.\(9\) b.\(5\) c.\(8\)

3. a.\(2\) b.\(4\) c.\(5\)

5. a.\(-6\) b.\(-6\) c.\(\frac{1}{6}\)

7. a. 不是真的 b.\(-3\) c.\(\frac{1}{3}\)

9. a. 不是真的 b.\(-6\) c.\(\frac{1}{6}\)

11. a. 不是真的 b.\(-10\) c.\(\frac{1}{10}\)

练习集 D：使用以下方法简化表达式\(a^{\frac{m}{n}}\)

在以下练习中，用有理指数书写。

1. a.\(\sqrt{m^{5}}\) b.\((\sqrt[3]{3 y})^{7}\) c.\(\sqrt[5]{\left(\dfrac{4 x}{5 y}\right)^{3}}\)
2. a.\(\sqrt[4]{r^{7}}\) b.\((\sqrt[5]{2 p q})^{3}\) c.\(\sqrt[4]{\left(\dfrac{12 m}{7 n}\right)^{3}}\)
3. a.\(\sqrt[5]{u^{2}}\) b.\((\sqrt[3]{6 x})^{5}\) c.\(\sqrt[4]{\left(\dfrac{18 a}{5 b}\right)^{7}}\)
4. a.\(\sqrt[3]{a}\) b.\((\sqrt[4]{21 v})^{3}\) c.\(\sqrt[4]{\left(\dfrac{2 x y}{5 z}\right)^{2}}\)

回答

1. a.\(m^{\frac{5}{2}}\) b.\((3 y)^{\frac{7}{3}}\) c.\(\left(\dfrac{4 x}{5 y}\right)^{\frac{3}{5}}\)

3. a.\(u^{\frac{2}{5}}\) b.\((6 x)^{\frac{5}{3}}\) c.\(\left(\dfrac{18 a}{5 b}\right)^{\frac{7}{4}}\)

练习集 E：使用以下方法简化表达式\(a^{\frac{m}{n}}\)

在以下练习中，进行简化。

1. a.\(64^{\frac{5}{2}}\) b.\(81^{\frac{-3}{2}}\) c.\((-27)^{\frac{2}{3}}\)
2. a.\(25^{\frac{3}{2}}\) b.\(9^{-\frac{3}{2}}\) c.\((-64)^{\frac{2}{3}}\)
3. a.\(32^{\frac{2}{5}}\) b.\(27^{-\frac{2}{3}}\) c.\((-25)^{\frac{1}{2}}\)
4. a.\(100^{\frac{3}{2}}\) b.\(49^{-\frac{5}{2}}\) c.\((-100)^{\frac{3}{2}}\)
5. a.\(-9^{\frac{3}{2}}\) b.\(-9^{-\frac{3}{2}}\) c.\((-9)^{\frac{3}{2}}\)
6. a.\(-64^{\frac{3}{2}}\) b.\(-64^{-\frac{3}{2}}\) c.\((-64)^{\frac{3}{2}}\)

回答

1. a.\(32,768\) b.\(\frac{1}{729}\) c.\(9\)

3. a.\(4\) b\(\frac{1}{9}\). c. 不是真的

5. a.\(-27\) b\(-\frac{1}{27}\). c. 不是真的

练习 SET F：使用指数定律简化带有有理指数的表达式

在以下练习中，进行简化。 假设所有变量均为正。

1. a.\(c^{\frac{1}{4}} \cdot c^{\frac{5}{8}}\) b.\(\left(p^{12}\right)^{\frac{3}{4}}\) c.\(\dfrac{r^{\frac{4}{5}}}{r^{\frac{9}{5}}}\)
2. a.\(6^{\frac{5}{2}} \cdot 6^{\frac{1}{2}}\) b.\(\left(b^{15}\right)^{\frac{3}{5}}\) c.\(\dfrac{w^{\frac{2}{7}}}{w^{\frac{9}{7}}}\)
3. a.\(y^{\frac{1}{2}} \cdot y^{\frac{3}{4}}\) b.\(\left(x^{12}\right)^{\frac{2}{3}}\) c.\(\dfrac{m^{\frac{5}{8}}}{m^{\frac{13}{8}}}\)
4. a.\(q^{\frac{2}{3}} \cdot q^{\frac{5}{6}}\) b.\(\left(h^{6}\right)^{\frac{4}{3}}\) c.\(\dfrac{n^{\frac{3}{5}}}{n^{\frac{8}{5}}}\)
5. a.\(\left(27 q^{\frac{3}{2}}\right)^{\frac{4}{3}}\) b。\(\left(a^{\frac{1}{3}} b^{\frac{2}{3}}\right)^{\frac{3}{2}}\)
6. a.\(\left(64 s^{\frac{3}{7}}\right)^{\frac{1}{6}}\) b。\(\left(m^{\frac{4}{3}} n^{\frac{1}{2}}\right)^{\frac{3}{4}}\)
7. a.\(\left(16 u^{\frac{1}{3}}\right)^{\frac{3}{4}}\) b。\(\left(4 p^{\frac{1}{3}} q^{\frac{1}{2}}\right)^{\frac{3}{2}}\)
8. a.\(\left(625 n^{\frac{8}{3}}\right)^{\frac{3}{4}}\) b。\(\left(9 x^{\frac{2}{5}} y^{\frac{3}{5}}\right)^{\frac{5}{2}}\)
9. a.\(\dfrac{r^{\frac{5}{2}} \cdot r^{-\frac{1}{2}}}{r^{-\frac{3}{2}}}\) b。\(\left(\dfrac{36 s^{\frac{1}{5}} t^{-\frac{3}{2}}}{s^{-\frac{9}{5}} t^{\frac{1}{2}}}\right)^{\frac{1}{2}}\)
10. a.\(\dfrac{a^{\frac{3}{4}} \cdot a^{-\frac{1}{4}}}{a^{-\frac{10}{4}}}\) b。\(\left(\dfrac{27 b^{\frac{2}{3}} c^{-\frac{5}{2}}}{b^{-\frac{7}{3}} c^{\frac{1}{2}}}\right)^{\frac{1}{3}}\)
11. a.\(\dfrac{c^{\frac{5}{3}} \cdot c^{-\frac{1}{3}}}{c^{-\frac{2}{3}}}\) b。\(\left(\dfrac{8 x^{\frac{5}{3}} y^{-\frac{1}{2}}}{27 x^{-\frac{4}{3}} y^{\frac{5}{2}}}\right)^{\frac{1}{3}}\)
12. a.\(\dfrac{m^{\frac{7}{4}} \cdot m^{-\frac{5}{4}}}{m^{-\frac{2}{4}}}\) b。\(\left(\dfrac{16 m^{\frac{1}{5}} n^{\frac{3}{2}}}{81 m^{\frac{9}{5}} n^{-\frac{1}{2}}}\right)^{\frac{1}{4}}\)

回答

1. a.\(c^{\frac{7}{8}}\) b.\(p^{9}\) c.\(\frac{1}{r}\)

3. a.\(y^{\frac{5}{4}}\) b.\(x^{8}\) c.\(\dfrac{1}{m}\)

5. a.\(81 q^{2}\) b.\(a^{\frac{1}{2}} b\)

7. a.\(8 u^{\frac{1}{4}}\) b.\(8 p^{\frac{1}{2}} q^{\frac{3}{4}}\)

9. a.\(r^{\frac{7}{2}}\) b.\(\dfrac{6 s}{t}\)

11. a.\(c^{2}\) b.\(\dfrac{2x}{3y}\)

练习套装 G：写作练习

1. 显示两种不同的代数方法以进行简化\(4^{\frac{3}{2}}\)。 解释你的所有步骤。
2. 解释为什么\((-16)^{\frac{3}{2}}\)无法计算表达式。

回答

1。 答案会有所不同。

自检

a. 完成练习后，使用此清单评估您对本节目标的掌握程度。

图 8.3.4

b. 关于你对本部分的掌握程度，这份清单告诉了你什么？ 你会采取哪些措施来改进？