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8.6E：练习

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    练习成就完美

    练习集 A：除以平方根

    在以下练习中，进行简化。

    1. a.\(\dfrac{\sqrt{128}}{\sqrt{72}}\quad\) b.\(\dfrac{\sqrt[3]{128}}{\sqrt[3]{54}}\)

    2. a.\(\dfrac{\sqrt{48}}{\sqrt{75}}\quad\) b.\(\dfrac{\sqrt[3]{81}}{\sqrt[3]{24}}\)

    3. a.\(\dfrac{\sqrt{200 m^{5}}}{\sqrt{98 m}}\quad\) b.\(\dfrac{\sqrt[3]{54 y^{2}}}{\sqrt[3]{2 y^{5}}}\)

    4. a.\(\dfrac{\sqrt{108 n^{7}}}{\sqrt{243 n^{3}}}\quad\) b.\(\dfrac{\sqrt[3]{54 y}}{\sqrt[3]{16 y^{4}}}\)

    5. a.\(\dfrac{\sqrt{75 r^{3}}}{\sqrt{108 r^{7}}}\quad\) b.\(\dfrac{\sqrt[3]{24 x^{7}}}{\sqrt[3]{81 x^{4}}}\)

    6. a.\(\dfrac{\sqrt{196 q}}{\sqrt{484 q^{5}}}\quad\) b.\(\dfrac{\sqrt[3]{16 m^{4}}}{\sqrt[3]{54 m}}\)

    7. a.\(\dfrac{\sqrt{108 p^{5} q^{2}}}{\sqrt{3 p^{3} q^{6}}}\quad\) b.\(\dfrac{\sqrt[3]{-16 a^{4} b^{-2}}}{\sqrt[3]{2 a^{-2} b}}\)

    8. a.\(\dfrac{\sqrt{98 r s^{10}}}{\sqrt{2 r^{3} s^{4}}}\quad\) b.\(\dfrac{\sqrt[3]{-375 y^{4} z^{2}}}{\sqrt[3]{3 y^{-2} z^{4}}}\)

    9. a.\(\dfrac{\sqrt{320 m n^{-5}}}{\sqrt{45 m^{-7} n^{3}}}\quad\) b.\(\dfrac{\sqrt[3]{16 x^{4} y^{-2}}}{\sqrt[3]{-54 x^{-2} y^{4}}}\)

    10. a.\(\dfrac{\sqrt{810 c^{-3} d^{7}}}{\sqrt{1000 c d}}\quad\) b.\(\dfrac{\sqrt[3]{24 a^{7} b^{-1}}}{\sqrt[3]{-81 a^{-2} b^{2}}}\)

    11。 \(\dfrac{\sqrt{56 x^{5} y^{4}}}{\sqrt{2 x y^{3}}}\)

    12。 \(\dfrac{\sqrt{72 a^{3} b^{6}}}{\sqrt{3 a b^{3}}}\)

    13。 \(\dfrac{\sqrt[3]{48 a^{3} b^{6}}}{\sqrt[3]{3 a^{-1} b^{3}}}\)

    14。 \(\dfrac{\sqrt[3]{162 x^{-3} y^{6}}}{\sqrt[3]{2 x^{3} y^{-2}}}\)

    回答

    1. a.\(\dfrac{4}{3}\) b.\(\dfrac{4}{3}\)

    3. a.\(\dfrac{10 m^{2}}{7}\) b.\(\dfrac{3}{y}\)

    5. a.\(\dfrac{5}{6 r^{2}}\) b.\(\dfrac{2x}{3}\)

    7. a.\(\dfrac{6 p}{q^{2}}\) b.\(-\dfrac{2 a^{2}}{b}\)

    9. a.\(\dfrac{8 m^{4}}{3 n^{4}}\) b.\(-\dfrac{2 x^{2}}{3 y^{2}}\)

    11。 \(4 x^{4} \sqrt{7 y}\)

    13。 \(2 a b \sqrt[3]{2 a}\)

    练习集 B：合理化单项分母

    在以下练习中，合理化分母。

    15. a.\(\dfrac{10}{\sqrt{6}}\quad\) b.\(\sqrt{\dfrac{4}{27}}\quad\) c.\(\dfrac{10}{\sqrt{5 x}}\)

    16. a.\(\dfrac{8}{\sqrt{3}}\quad\) b.\(\sqrt{\dfrac{7}{40}}\quad\) c.\(\dfrac{8}{\sqrt{2 y}}\)

    17. a.\(\dfrac{6}{\sqrt{7}}\quad\) b.\(\sqrt{\dfrac{8}{45}}\quad\) c.\(\dfrac{12}{\sqrt{3 p}}\)

    18. a.\(\dfrac{4}{\sqrt{5}}\quad\) b.\(\sqrt{\dfrac{27}{80}}\quad\) c.\(\dfrac{18}{\sqrt{6 q}}\)

    19. a.\(\dfrac{1}{\sqrt[3]{5}}\quad\) b.\(\sqrt[3]{\dfrac{5}{24}}\quad\) c.\(\dfrac{4}{\sqrt[3]{36 a}}\)

    20. a.\(\dfrac{1}{\sqrt[3]{3}}\quad\) b.\(\sqrt[3]{\dfrac{5}{32}}\quad\) c.\(\dfrac{7}{\sqrt[3]{49 b}}\)

    21. a.\(\dfrac{1}{\sqrt[3]{11}}\quad\) b.\(\sqrt[3]{\dfrac{7}{54}}\quad\) c.\(\dfrac{3}{\sqrt[3]{3 x^{2}}}\)

    22. a.\(\dfrac{1}{\sqrt[3]{13}}\quad\) b.\(\sqrt[3]{\dfrac{3}{128}}\quad\) c.\(\dfrac{3}{\sqrt[3]{6 y^{2}}}\)

    23. a.\(\dfrac{1}{\sqrt[4]{7}}\quad\) b.\(\sqrt[4]{\dfrac{5}{32}}\quad\) c.\(\dfrac{4}{\sqrt[4]{4 x^{2}}}\)

    24. a.\(\dfrac{1}{\sqrt[4]{4}}\quad\) b.\(\sqrt[4]{\dfrac{9}{32}}\quad\) c.\(\dfrac{6}{\sqrt[4]{9 x^{3}}}\)

    25. a.\(\dfrac{1}{\sqrt[4]{9}}\quad\) b.\(\sqrt[4]{\dfrac{25}{128}}\quad\) c.\(\dfrac{6}{\sqrt[4]{27 a}}\)

    26. a.\(\dfrac{1}{\sqrt[4]{8}}\quad\) b.\(\sqrt[4]{\dfrac{27}{128}}\quad\) c.\(\dfrac{16}{\sqrt[4]{64 b^{2}}}\)

    回答

    15. a.\(\dfrac{5 \sqrt{6}}{3}\) b.\(\dfrac{2 \sqrt{3}}{9}\) c.\(\dfrac{2 \sqrt{5 x}}{x}\)

    17. a.\(\dfrac{6 \sqrt{7}}{7}\) b.\(\dfrac{2 \sqrt{10}}{15}\) c.\(\dfrac{4 \sqrt{3 p}}{p}\)

    19. a.\(\dfrac{\sqrt[3]{25}}{5}\) b.\(\dfrac{\sqrt[3]{45}}{6}\) c.\(\dfrac{2 \sqrt[3]{6 a^{2}}}{3 a}\)

    21. a.\(\dfrac{\sqrt[3]{121}}{11}\) b.\(\dfrac{\sqrt[3]{28}}{6}\) c.\(\dfrac{\sqrt[3]{9 x}}{x}\)

    23. a.\(\dfrac{\sqrt[4]{343}}{7}\) b.\(\dfrac{\sqrt[4]{40}}{4}\) c.\(\dfrac{2 \sqrt[4]{4 x^{2}}}{x}\)

    25. a.\(\dfrac{\sqrt[4]{9}}{3}\) b.\(\dfrac{\sqrt[4]{50}}{4}\) c.\(\dfrac{2 \sqrt[4]{3 a^{2}}}{a}\)

    练习集 C：合理化两项分母

    在以下练习中，进行简化。

    27。 \(\dfrac{8}{1-\sqrt{5}}\)

    28。 \(\dfrac{7}{2-\sqrt{6}}\)

    29。 \(\dfrac{6}{3-\sqrt{7}}\)

    30。 \(\dfrac{5}{4-\sqrt{11}}\)

    31。 \(\dfrac{\sqrt{3}}{\sqrt{m}-\sqrt{5}}\)

    32。 \(\dfrac{\sqrt{5}}{\sqrt{n}-\sqrt{7}}\)

    33。 \(\dfrac{\sqrt{2}}{\sqrt{x}-\sqrt{6}}\)

    34。 \(\dfrac{\sqrt{7}}{\sqrt{y}+\sqrt{3}}\)

    35。 \(\dfrac{\sqrt{r}+\sqrt{5}}{\sqrt{r}-\sqrt{5}}\)

    36。 \(\dfrac{\sqrt{s}-\sqrt{6}}{\sqrt{s}+\sqrt{6}}\)

    37。 \(\dfrac{\sqrt{x}+\sqrt{8}}{\sqrt{x}-\sqrt{8}}\)

    38。 \(\dfrac{\sqrt{m}-\sqrt{3}}{\sqrt{m}+\sqrt{3}}\)

    回答

    27。 \(-2(1+\sqrt{5})\)

    29。 \(3(3+\sqrt{7})\)

    31。 \(\dfrac{\sqrt{3}(\sqrt{m}+\sqrt{5})}{m-5}\)

    33。 \(\dfrac{\sqrt{2}(\sqrt{x}+\sqrt{6})}{x-6}\)

    35。 \(\dfrac{(\sqrt{r}+\sqrt{5})^{2}}{r-5}\)

    37。 \(\dfrac{(\sqrt{x}+2 \sqrt{2})^{2}}{x-8}\)

    练习套装 D：写作练习
      1. 简化\(\sqrt{\dfrac{27}{3}}\)并解释您的所有步骤。
      2. 简化\(\sqrt{\dfrac{27}{5}}\)并解释您的所有步骤。
      3. 为什么两种简化平方根的方法不同？
    2. 解释 “合理化分母” 这句话中的 “合理化” 一词的含义。
    3. 解释为什么\(\sqrt{2x}-3\)乘以其共轭会得到没有自由基的表达式。
    4. 解释为什么\(\dfrac{7}{\sqrt[3]{x}}\)乘以\(\dfrac{\sqrt[3]{x}}{\sqrt[3]{x}}\)不能使分母合理化。
    回答

    1。 答案会有所不同

    3。 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。

    此表有 4 行和 4 列。 第一行是标题行，它标记了每列。 第一列标题是 “我可以”，第二个是 “自信”，第三栏是 “有帮助”，第四列是 “不，我不明白”。™ 第一列下方是 “划分激进表达式” 这句话。“合理化一个单词分母” 和 “合理化两个术语的分母”。 其他栏留空，以便学员可以指明他们对每个主题的掌握程度。
    图 8.5.63

    b. 看完清单后，你认为你为下一节做好了充分的准备吗？ 为什么或者为什么不呢？

    This page titled 8.6E：练习 is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by OpenStax via source content that was edited to the style and standards of the LibreTexts platform.