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5.9: 有理指数

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    指数并不总是整数。 本节将探讨指数是有理数的情况。 当指数是有理数时，表达式可以写成带有激进的表达式。 规则是用与原始问题相同的形式写出答案（如果你以指数开头，以指数结尾，或者如果你以激进开头，以激进结尾）。

    定义：表单的有理指数\(\dfrac{1}{n}\)

    对于任何实数\(a\)和任何整数\(n\)，指数为的表达式\(\dfrac{1}{n}\)可以表示为如下所示

    \[a^{\frac{1}{n}} = \sqrt[n]{a} \nonumber \]

    注意：\(n\)是激进中的索引。 \(\sqrt[n]{a}\)被读为 “a 的第 n 个根”

    注意：当激进没有可见索引时，默认情况下，索引为\(2\)（平方根）。 大于的指数\(2\)将在激进部分上标出。

    示例 Template:Index
    1. \((4)^{\frac{1}{2}} = \sqrt{4} = 2\)\(\text{Index is \(2\)默认}\)
    2. \( (x)^{\frac{1}{7}} = \sqrt[7]{x}\)\(\text{Index is \(7\)}\)
    3. \((−3y)^{\frac{1}{3}} = \sqrt[3]{(-3y)}\)\(\text{Index is \(3\)}\)

    现在，让我们观察一下当指数是带有分子的有理数时会发生什么\(\neq 1\)。

    定义：表单的有理指数\(\dfrac{m}{n}\)

    对于任何实数\(a\)和任何整数\(n\)和\(m\)，指数为的表达式\(\dfrac{m}{n}\)可以表示为如下所示

    \[a^{\frac{m}{n}} = \sqrt[n]{a^m} \text{ or } (\sqrt[n]{a})^m \nonumber \]

    注意：\(n\)是激进中的索引，\(m\)是基数的力量。

    示例 Template:Index

    用激进的形式写下以下内容

    1. \((x)^{\frac{2}{3}} = \sqrt[3]{x^2} = (\sqrt[3]{x})^2\)\(\text{Index is \(3\)并将 base 提升到\(2\)。}\)
    2. \((5t)^{\frac{7}{8}} = \sqrt[8]{5t^7} = (\sqrt[8]{5t})^7\)\(\text{Index is \(8\)然后 base 被提升到\(7\)力量。}\)
    3. \((x)^{\frac{2}{3}} = \sqrt[3]{x^2} = (\sqrt[3]{x})^2\)\(\text{Index is \(3\)然后将基地提升到权力\(2\)。}\)
    4. \(\begin{array} &&(z)^{−\frac{5}{9}} &\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\text{Given} \\ &= \dfrac{1}{(z)^{\frac{5}{9}}} &\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\text{Negative exponent rule applied} \\ &= \dfrac{1}{\sqrt[9]{x^5}} \text{ or } \left( \dfrac{1}{\sqrt[9]{x}} \right)^5 &\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\text{Rational exponent written as a radical.} \end{array}\)
    5. \(\left( \dfrac{3}{4} \right)^{\frac{5}{7}} = \sqrt[7]{\dfrac{3}{4}^5}\)\(\text{Rational exponent written as radical with index \(7\)然后 base 提升到\(5\)了。}\)
    练习 Template:Index

    用激进的形式写下以下内容。

    1. \((x)^{\frac{5}{7}}\)
    2. \((xy)^{\frac{9}{8}}\)
    3. \((x)^{\frac{9}{5}}\)
    4. \((z)^{−\frac{11}{13}}\)
    5. \(\left( \dfrac{x}{4} \right)^{\frac{6}{9}}\)
    6. \(6(y)^{\frac{1}{17}}\)
    7. \((6y)^{\frac{1}{17}}\)
    8. \(\left( \dfrac{3}{4} \right)^{\frac{x}{y}}\)
    9. \(\left( \dfrac{7}{4} \right)^{(−\frac{x}{y})}\)
    Template:AutoNumHeadings

    This page titled 5.9: 有理指数 is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Victoria Dominguez, Cristian Martinez, & Sanaa Saykali (ASCCC Open Educational Resources Initiative) .