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5.9: Expoentes racionais

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    Os expoentes nem sempre são números inteiros. Esta seção examinará os casos em que um expoente é um número racional. Quando um expoente é um número racional, a expressão pode ser escrita como uma expressão com um radical. A regra é escrever sua resposta na mesma forma do problema original (se você começar com expoentes, terminar com expoentes ou se começar com radicais, terminar com radicais).

    Definição: Expoentes racionais da forma\(\dfrac{1}{n}\)

    Para qualquer número real\(a\) e qualquer número inteiro\(n\), uma expressão com o expoente de\(\dfrac{1}{n}\) pode ser expressa da seguinte forma

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

    Nota:\(n\) é o índice no radical. \(\sqrt[n]{a}\)é lido como “a enésima raiz de a”

    Nota: Quando o radical não tem um índice visível, por padrão, o índice é\(2\) (raiz quadrada). Índices maiores do que\(2\) serão marcados no radical.

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

    Agora, vamos observar o que acontece quando o expoente é um número racional com numerador\(\neq 1\).

    Definição: Expoentes racionais da forma\(\dfrac{m}{n}\)

    Para qualquer número real\(a\) e qualquer número inteiro\(n\) e\(m\), uma expressão com o expoente de\(\dfrac{m}{n}\) pode ser expressa da seguinte forma

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

    Nota:\(n\) é o índice no radical e\(m\) é a potência da base.

    Exemplo Template:index

    Escreva o seguinte em forma radical

    1. \((x)^{\frac{2}{3}} = \sqrt[3]{x^2} = (\sqrt[3]{x})^2\)\(\text{Index is \(3\)e a base é elevada à potência de\(2\).}\)
    2. \((5t)^{\frac{7}{8}} = \sqrt[8]{5t^7} = (\sqrt[8]{5t})^7\)\(\text{Index is \(8\)e a base é elevada à\(7\) potência.}\)
    3. \((x)^{\frac{2}{3}} = \sqrt[3]{x^2} = (\sqrt[3]{x})^2\)\(\text{Index is \(3\)e base elevada à potência\(2\).}\)
    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 \(7\)e base elevada à potência de\(5\).}\)
    Exercício Template:index

    Escreva o seguinte de forma radical.

    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: Expoentes racionais is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Victoria Dominguez, Cristian Martinez, and Sanaa Saykali (ASCCC Open Educational Resources Initiative) .