代数示例

$f (x) = {(x^{3} - 2)}^{\frac{1}{5}} + 3$

解题步骤 1

将 $f (x) = {(x^{3} - 2)}^{\frac{1}{5}} + 3$ 写为等式。

$y = {(x^{3} - 2)}^{\frac{1}{5}} + 3$

解题步骤 2

交换变量。

$x = {(y^{3} - 2)}^{\frac{1}{5}} + 3$

解题步骤 3

求解 $y$ 。

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解题步骤 3.1

将方程重写为 ${(y^{3} - 2)}^{\frac{1}{5}} + 3 = x$ 。

${(y^{3} - 2)}^{\frac{1}{5}} + 3 = x$

解题步骤 3.2

从等式两边同时减去 $3$ 。

${(y^{3} - 2)}^{\frac{1}{5}} = x - 3$

解题步骤 3.3

将方程两边同时进行 $5$ 次方运算以消去左边的分数指数。

${({(y^{3} - 2)}^{\frac{1}{5}})}^{5} = {(x - 3)}^{5}$

解题步骤 3.4

化简指数。

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解题步骤 3.4.1

化简左边。

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解题步骤 3.4.1.1

化简 ${({(y^{3} - 2)}^{\frac{1}{5}})}^{5}$ 。

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解题步骤 3.4.1.1.1

将 ${({(y^{3} - 2)}^{\frac{1}{5}})}^{5}$ 中的指数相乘。

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解题步骤 3.4.1.1.1.1

运用幂法则并将指数相乘， ${(a^{m})}^{n} = a^{m n}$ 。

${(y^{3} - 2)}^{\frac{1}{5} \cdot 5} = {(x - 3)}^{5}$

解题步骤 3.4.1.1.1.2

约去 $5$ 的公因数。

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解题步骤 3.4.1.1.1.2.1

约去公因数。

${(y^{3} - 2)}^{\frac{1}{5} \cdot 5} = {(x - 3)}^{5}$

解题步骤 3.4.1.1.1.2.2

重写表达式。

${(y^{3} - 2)}^{1} = {(x - 3)}^{5}$

解题步骤 3.4.1.1.2

化简。

$y^{3} - 2 = {(x - 3)}^{5}$

解题步骤 3.4.2

化简右边。

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解题步骤 3.4.2.1

化简 ${(x - 3)}^{5}$ 。

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解题步骤 3.4.2.1.1

使用二项式定理。

$y^{3} - 2 = x^{5} + 5 x^{4} \cdot - 3 + 10 x^{3} {(- 3)}^{2} + 10 x^{2} {(- 3)}^{3} + 5 x {(- 3)}^{4} + {(- 3)}^{5}$

解题步骤 3.4.2.1.2

化简每一项。

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解题步骤 3.4.2.1.2.1

将 $- 3$ 乘以 $5$ 。

$y^{3} - 2 = x^{5} - 15 x^{4} + 10 x^{3} {(- 3)}^{2} + 10 x^{2} {(- 3)}^{3} + 5 x {(- 3)}^{4} + {(- 3)}^{5}$

解题步骤 3.4.2.1.2.2

对 $- 3$ 进行 $2$ 次方运算。

$y^{3} - 2 = x^{5} - 15 x^{4} + 10 x^{3} \cdot 9 + 10 x^{2} {(- 3)}^{3} + 5 x {(- 3)}^{4} + {(- 3)}^{5}$

解题步骤 3.4.2.1.2.3

将 $9$ 乘以 $10$ 。

$y^{3} - 2 = x^{5} - 15 x^{4} + 90 x^{3} + 10 x^{2} {(- 3)}^{3} + 5 x {(- 3)}^{4} + {(- 3)}^{5}$

解题步骤 3.4.2.1.2.4

对 $- 3$ 进行 $3$ 次方运算。

$y^{3} - 2 = x^{5} - 15 x^{4} + 90 x^{3} + 10 x^{2} \cdot - 27 + 5 x {(- 3)}^{4} + {(- 3)}^{5}$

解题步骤 3.4.2.1.2.5

将 $- 27$ 乘以 $10$ 。

$y^{3} - 2 = x^{5} - 15 x^{4} + 90 x^{3} - 270 x^{2} + 5 x {(- 3)}^{4} + {(- 3)}^{5}$

解题步骤 3.4.2.1.2.6

对 $- 3$ 进行 $4$ 次方运算。

$y^{3} - 2 = x^{5} - 15 x^{4} + 90 x^{3} - 270 x^{2} + 5 x \cdot 81 + {(- 3)}^{5}$

解题步骤 3.4.2.1.2.7

将 $81$ 乘以 $5$ 。

$y^{3} - 2 = x^{5} - 15 x^{4} + 90 x^{3} - 270 x^{2} + 405 x + {(- 3)}^{5}$

解题步骤 3.4.2.1.2.8

对 $- 3$ 进行 $5$ 次方运算。

$y^{3} - 2 = x^{5} - 15 x^{4} + 90 x^{3} - 270 x^{2} + 405 x - 243$

解题步骤 3.5

求解 $y$ 。

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解题步骤 3.5.1

将所有不包含 $y$ 的项移到等式右边。

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解题步骤 3.5.1.1

在等式两边都加上 $2$ 。

$y^{3} = x^{5} - 15 x^{4} + 90 x^{3} - 270 x^{2} + 405 x - 243 + 2$

解题步骤 3.5.1.2

将 $- 243$ 和 $2$ 相加。

$y^{3} = x^{5} - 15 x^{4} + 90 x^{3} - 270 x^{2} + 405 x - 241$

解题步骤 3.5.2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$y = \sqrt[3]{x^{5} - 15 x^{4} + 90 x^{3} - 270 x^{2} + 405 x - 241}$

解题步骤 4

Replace $y$ with $f^{- 1} (x)$ to show the final answer.

$f^{- 1} (x) = \sqrt[3]{x^{5} - 15 x^{4} + 90 x^{3} - 270 x^{2} + 405 x - 241}$

解题步骤 5

验证 $f^{- 1} (x) = \sqrt[3]{x^{5} - 15 x^{4} + 90 x^{3} - 270 x^{2} + 405 x - 241}$ 是否为 $f (x) = {(x^{3} - 2)}^{\frac{1}{5}} + 3$ 的反函数。

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解题步骤 5.1

要验证反函数，请检查 $f^{- 1} (f (x)) = x$ 和 $f (f^{- 1} (x)) = x$ 是否成立。

解题步骤 5.2

计算 $f^{- 1} (f (x))$ 。

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解题步骤 5.2.1

建立复合结果函数。

$f^{- 1} (f (x))$

解题步骤 5.2.2

通过将 $f$ 的值代入 $f^{- 1}$ 来计算 $f^{- 1} ({(x^{3} - 2)}^{\frac{1}{5}} + 3)$ 。

$f^{- 1} ({(x^{3} - 2)}^{\frac{1}{5}} + 3) = \sqrt[3]{{({(x^{3} - 2)}^{\frac{1}{5}} + 3)}^{5} - 15 {({(x^{3} - 2)}^{\frac{1}{5}} + 3)}^{4} + 90 {({(x^{3} - 2)}^{\frac{1}{5}} + 3)}^{3} - 270 {({(x^{3} - 2)}^{\frac{1}{5}} + 3)}^{2} + 405 ({(x^{3} - 2)}^{\frac{1}{5}} + 3) - 241}$

解题步骤 5.2.3

使用二项式定理。

$f^{- 1} ({(x^{3} - 2)}^{\frac{1}{5}} + 3) = \sqrt[3]{{({(x^{3} - 2)}^{\frac{1}{5}})}^{5} + 5 {({(x^{3} - 2)}^{\frac{1}{5}})}^{4} \cdot 3 + 10 {({(x^{3} - 2)}^{\frac{1}{5}})}^{3} \cdot 3^{2} + 10 {({(x^{3} - 2)}^{\frac{1}{5}})}^{2} \cdot 3^{3} + 5 {(x^{3} - 2)}^{\frac{1}{5}} \cdot 3^{4} + 3^{5} - 15 {({(x^{3} - 2)}^{\frac{1}{5}} + 3)}^{4} + 90 {({(x^{3} - 2)}^{\frac{1}{5}} + 3)}^{3} - 270 {({(x^{3} - 2)}^{\frac{1}{5}} + 3)}^{2} + 405 ({(x^{3} - 2)}^{\frac{1}{5}} + 3) - 241}$

解题步骤 5.2.4

化简项。

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解题步骤 5.2.4.1

化简每一项。

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解题步骤 5.2.4.1.1

将 ${({(x^{3} - 2)}^{\frac{1}{5}})}^{5}$ 中的指数相乘。

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解题步骤 5.2.4.1.1.1

运用幂法则并将指数相乘， ${(a^{m})}^{n} = a^{m n}$ 。

$f^{- 1} ({(x^{3} - 2)}^{\frac{1}{5}} + 3) = \sqrt[3]{{(x^{3} - 2)}^{\frac{1}{5} \cdot 5} + 5 {({(x^{3} - 2)}^{\frac{1}{5}})}^{4} \cdot 3 + 10 {({(x^{3} - 2)}^{\frac{1}{5}})}^{3} \cdot 3^{2} + 10 {({(x^{3} - 2)}^{\frac{1}{5}})}^{2} \cdot 3^{3} + 5 {(x^{3} - 2)}^{\frac{1}{5}} \cdot 3^{4} + 3^{5} - 15 {({(x^{3} - 2)}^{\frac{1}{5}} + 3)}^{4} + 90 {({(x^{3} - 2)}^{\frac{1}{5}} + 3)}^{3} - 270 {({(x^{3} - 2)}^{\frac{1}{5}} + 3)}^{2} + 405 ({(x^{3} - 2)}^{\frac{1}{5}} + 3) - 241}$

解题步骤 5.2.4.1.1.2

约去 $5$ 的公因数。

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解题步骤 5.2.4.1.1.2.1

约去公因数。

解题步骤 5.2.4.1.1.2.2

重写表达式。

$f^{- 1} ({(x^{3} - 2)}^{\frac{1}{5}} + 3) = \sqrt[3]{(x^{3} - 2) + 5 {({(x^{3} - 2)}^{\frac{1}{5}})}^{4} \cdot 3 + 10 {({(x^{3} - 2)}^{\frac{1}{5}})}^{3} \cdot 3^{2} + 10 {({(x^{3} - 2)}^{\frac{1}{5}})}^{2} \cdot 3^{3} + 5 {(x^{3} - 2)}^{\frac{1}{5}} \cdot 3^{4} + 3^{5} - 15 {({(x^{3} - 2)}^{\frac{1}{5}} + 3)}^{4} + 90 {({(x^{3} - 2)}^{\frac{1}{5}} + 3)}^{3} - 270 {({(x^{3} - 2)}^{\frac{1}{5}} + 3)}^{2} + 405 ({(x^{3} - 2)}^{\frac{1}{5}} + 3) - 241}$

解题步骤 5.2.4.1.2

化简。

$f^{- 1} ({(x^{3} - 2)}^{\frac{1}{5}} + 3) = \sqrt[3]{x^{3} - 2 + 5 {({(x^{3} - 2)}^{\frac{1}{5}})}^{4} \cdot 3 + 10 {({(x^{3} - 2)}^{\frac{1}{5}})}^{3} \cdot 3^{2} + 10 {({(x^{3} - 2)}^{\frac{1}{5}})}^{2} \cdot 3^{3} + 5 {(x^{3} - 2)}^{\frac{1}{5}} \cdot 3^{4} + 3^{5} - 15 {({(x^{3} - 2)}^{\frac{1}{5}} + 3)}^{4} + 90 {({(x^{3} - 2)}^{\frac{1}{5}} + 3)}^{3} - 270 {({(x^{3} - 2)}^{\frac{1}{5}} + 3)}^{2} + 405 ({(x^{3} - 2)}^{\frac{1}{5}} + 3) - 241}$

解题步骤 5.2.4.1.3

将 ${({(x^{3} - 2)}^{\frac{1}{5}})}^{4}$ 中的指数相乘。

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解题步骤 5.2.4.1.3.1

运用幂法则并将指数相乘， ${(a^{m})}^{n} = a^{m n}$ 。

$f^{- 1} ({(x^{3} - 2)}^{\frac{1}{5}} + 3) = \sqrt[3]{x^{3} - 2 + 5 {(x^{3} - 2)}^{\frac{1}{5} \cdot 4} \cdot 3 + 10 {({(x^{3} - 2)}^{\frac{1}{5}})}^{3} \cdot 3^{2} + 10 {({(x^{3} - 2)}^{\frac{1}{5}})}^{2} \cdot 3^{3} + 5 {(x^{3} - 2)}^{\frac{1}{5}} \cdot 3^{4} + 3^{5} - 15 {({(x^{3} - 2)}^{\frac{1}{5}} + 3)}^{4} + 90 {({(x^{3} - 2)}^{\frac{1}{5}} + 3)}^{3} - 270 {({(x^{3} - 2)}^{\frac{1}{5}} + 3)}^{2} + 405 ({(x^{3} - 2)}^{\frac{1}{5}} + 3) - 241}$

解题步骤 5.2.4.1.3.2

组合 $\frac{1}{5}$ 和 $4$ 。

$f^{- 1} ({(x^{3} - 2)}^{\frac{1}{5}} + 3) = \sqrt[3]{x^{3} - 2 + 5 {(x^{3} - 2)}^{\frac{4}{5}} \cdot 3 + 10 {({(x^{3} - 2)}^{\frac{1}{5}})}^{3} \cdot 3^{2} + 10 {({(x^{3} - 2)}^{\frac{1}{5}})}^{2} \cdot 3^{3} + 5 {(x^{3} - 2)}^{\frac{1}{5}} \cdot 3^{4} + 3^{5} - 15 {({(x^{3} - 2)}^{\frac{1}{5}} + 3)}^{4} + 90 {({(x^{3} - 2)}^{\frac{1}{5}} + 3)}^{3} - 270 {({(x^{3} - 2)}^{\frac{1}{5}} + 3)}^{2} + 405 ({(x^{3} - 2)}^{\frac{1}{5}} + 3) - 241}$

解题步骤 5.2.4.1.4

将 $3$ 乘以 $5$ 。

$f^{- 1} ({(x^{3} - 2)}^{\frac{1}{5}} + 3) = \sqrt[3]{x^{3} - 2 + 15 {(x^{3} - 2)}^{\frac{4}{5}} + 10 {({(x^{3} - 2)}^{\frac{1}{5}})}^{3} \cdot 3^{2} + 10 {({(x^{3} - 2)}^{\frac{1}{5}})}^{2} \cdot 3^{3} + 5 {(x^{3} - 2)}^{\frac{1}{5}} \cdot 3^{4} + 3^{5} - 15 {({(x^{3} - 2)}^{\frac{1}{5}} + 3)}^{4} + 90 {({(x^{3} - 2)}^{\frac{1}{5}} + 3)}^{3} - 270 {({(x^{3} - 2)}^{\frac{1}{5}} + 3)}^{2} + 405 ({(x^{3} - 2)}^{\frac{1}{5}} + 3) - 241}$

解题步骤 5.2.4.1.5

将 ${({(x^{3} - 2)}^{\frac{1}{5}})}^{3}$ 中的指数相乘。

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解题步骤 5.2.4.1.5.1

运用幂法则并将指数相乘， ${(a^{m})}^{n} = a^{m n}$ 。

$f^{- 1} ({(x^{3} - 2)}^{\frac{1}{5}} + 3) = \sqrt[3]{x^{3} - 2 + 15 {(x^{3} - 2)}^{\frac{4}{5}} + 10 {(x^{3} - 2)}^{\frac{1}{5} \cdot 3} \cdot 3^{2} + 10 {({(x^{3} - 2)}^{\frac{1}{5}})}^{2} \cdot 3^{3} + 5 {(x^{3} - 2)}^{\frac{1}{5}} \cdot 3^{4} + 3^{5} - 15 {({(x^{3} - 2)}^{\frac{1}{5}} + 3)}^{4} + 90 {({(x^{3} - 2)}^{\frac{1}{5}} + 3)}^{3} - 270 {({(x^{3} - 2)}^{\frac{1}{5}} + 3)}^{2} + 405 ({(x^{3} - 2)}^{\frac{1}{5}} + 3) - 241}$

解题步骤 5.2.4.1.5.2

组合 $\frac{1}{5}$ 和 $3$ 。

$f^{- 1} ({(x^{3} - 2)}^{\frac{1}{5}} + 3) = \sqrt[3]{x^{3} - 2 + 15 {(x^{3} - 2)}^{\frac{4}{5}} + 10 {(x^{3} - 2)}^{\frac{3}{5}} \cdot 3^{2} + 10 {({(x^{3} - 2)}^{\frac{1}{5}})}^{2} \cdot 3^{3} + 5 {(x^{3} - 2)}^{\frac{1}{5}} \cdot 3^{4} + 3^{5} - 15 {({(x^{3} - 2)}^{\frac{1}{5}} + 3)}^{4} + 90 {({(x^{3} - 2)}^{\frac{1}{5}} + 3)}^{3} - 270 {({(x^{3} - 2)}^{\frac{1}{5}} + 3)}^{2} + 405 ({(x^{3} - 2)}^{\frac{1}{5}} + 3) - 241}$

解题步骤 5.2.4.1.6

对 $3$ 进行 $2$ 次方运算。

$f^{- 1} ({(x^{3} - 2)}^{\frac{1}{5}} + 3) = \sqrt[3]{x^{3} - 2 + 15 {(x^{3} - 2)}^{\frac{4}{5}} + 10 {(x^{3} - 2)}^{\frac{3}{5}} \cdot 9 + 10 {({(x^{3} - 2)}^{\frac{1}{5}})}^{2} \cdot 3^{3} + 5 {(x^{3} - 2)}^{\frac{1}{5}} \cdot 3^{4} + 3^{5} - 15 {({(x^{3} - 2)}^{\frac{1}{5}} + 3)}^{4} + 90 {({(x^{3} - 2)}^{\frac{1}{5}} + 3)}^{3} - 270 {({(x^{3} - 2)}^{\frac{1}{5}} + 3)}^{2} + 405 ({(x^{3} - 2)}^{\frac{1}{5}} + 3) - 241}$

解题步骤 5.2.4.1.7

将 $9$ 乘以 $10$ 。

$f^{- 1} ({(x^{3} - 2)}^{\frac{1}{5}} + 3) = \sqrt[3]{x^{3} - 2 + 15 {(x^{3} - 2)}^{\frac{4}{5}} + 90 {(x^{3} - 2)}^{\frac{3}{5}} + 10 {({(x^{3} - 2)}^{\frac{1}{5}})}^{2} \cdot 3^{3} + 5 {(x^{3} - 2)}^{\frac{1}{5}} \cdot 3^{4} + 3^{5} - 15 {({(x^{3} - 2)}^{\frac{1}{5}} + 3)}^{4} + 90 {({(x^{3} - 2)}^{\frac{1}{5}} + 3)}^{3} - 270 {({(x^{3} - 2)}^{\frac{1}{5}} + 3)}^{2} + 405 ({(x^{3} - 2)}^{\frac{1}{5}} + 3) - 241}$

解题步骤 5.2.4.1.8

将 ${({(x^{3} - 2)}^{\frac{1}{5}})}^{2}$ 中的指数相乘。

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解题步骤 5.2.4.1.8.1

运用幂法则并将指数相乘， ${(a^{m})}^{n} = a^{m n}$ 。

$f^{- 1} ({(x^{3} - 2)}^{\frac{1}{5}} + 3) = \sqrt[3]{x^{3} - 2 + 15 {(x^{3} - 2)}^{\frac{4}{5}} + 90 {(x^{3} - 2)}^{\frac{3}{5}} + 10 {(x^{3} - 2)}^{\frac{1}{5} \cdot 2} \cdot 3^{3} + 5 {(x^{3} - 2)}^{\frac{1}{5}} \cdot 3^{4} + 3^{5} - 15 {({(x^{3} - 2)}^{\frac{1}{5}} + 3)}^{4} + 90 {({(x^{3} - 2)}^{\frac{1}{5}} + 3)}^{3} - 270 {({(x^{3} - 2)}^{\frac{1}{5}} + 3)}^{2} + 405 ({(x^{3} - 2)}^{\frac{1}{5}} + 3) - 241}$

解题步骤 5.2.4.1.8.2

组合 $\frac{1}{5}$ 和 $2$ 。

$f^{- 1} ({(x^{3} - 2)}^{\frac{1}{5}} + 3) = \sqrt[3]{x^{3} - 2 + 15 {(x^{3} - 2)}^{\frac{4}{5}} + 90 {(x^{3} - 2)}^{\frac{3}{5}} + 10 {(x^{3} - 2)}^{\frac{2}{5}} \cdot 3^{3} + 5 {(x^{3} - 2)}^{\frac{1}{5}} \cdot 3^{4} + 3^{5} - 15 {({(x^{3} - 2)}^{\frac{1}{5}} + 3)}^{4} + 90 {({(x^{3} - 2)}^{\frac{1}{5}} + 3)}^{3} - 270 {({(x^{3} - 2)}^{\frac{1}{5}} + 3)}^{2} + 405 ({(x^{3} - 2)}^{\frac{1}{5}} + 3) - 241}$

解题步骤 5.2.4.1.9

对 $3$ 进行 $3$ 次方运算。

$f^{- 1} ({(x^{3} - 2)}^{\frac{1}{5}} + 3) = \sqrt[3]{x^{3} - 2 + 15 {(x^{3} - 2)}^{\frac{4}{5}} + 90 {(x^{3} - 2)}^{\frac{3}{5}} + 10 {(x^{3} - 2)}^{\frac{2}{5}} \cdot 27 + 5 {(x^{3} - 2)}^{\frac{1}{5}} \cdot 3^{4} + 3^{5} - 15 {({(x^{3} - 2)}^{\frac{1}{5}} + 3)}^{4} + 90 {({(x^{3} - 2)}^{\frac{1}{5}} + 3)}^{3} - 270 {({(x^{3} - 2)}^{\frac{1}{5}} + 3)}^{2} + 405 ({(x^{3} - 2)}^{\frac{1}{5}} + 3) - 241}$

解题步骤 5.2.4.1.10

将 $27$ 乘以 $10$ 。

$f^{- 1} ({(x^{3} - 2)}^{\frac{1}{5}} + 3) = \sqrt[3]{x^{3} - 2 + 15 {(x^{3} - 2)}^{\frac{4}{5}} + 90 {(x^{3} - 2)}^{\frac{3}{5}} + 270 {(x^{3} - 2)}^{\frac{2}{5}} + 5 {(x^{3} - 2)}^{\frac{1}{5}} \cdot 3^{4} + 3^{5} - 15 {({(x^{3} - 2)}^{\frac{1}{5}} + 3)}^{4} + 90 {({(x^{3} - 2)}^{\frac{1}{5}} + 3)}^{3} - 270 {({(x^{3} - 2)}^{\frac{1}{5}} + 3)}^{2} + 405 ({(x^{3} - 2)}^{\frac{1}{5}} + 3) - 241}$

解题步骤 5.2.4.1.11

对 $3$ 进行 $4$ 次方运算。

$f^{- 1} ({(x^{3} - 2)}^{\frac{1}{5}} + 3) = \sqrt[3]{x^{3} - 2 + 15 {(x^{3} - 2)}^{\frac{4}{5}} + 90 {(x^{3} - 2)}^{\frac{3}{5}} + 270 {(x^{3} - 2)}^{\frac{2}{5}} + 5 {(x^{3} - 2)}^{\frac{1}{5}} \cdot 81 + 3^{5} - 15 {({(x^{3} - 2)}^{\frac{1}{5}} + 3)}^{4} + 90 {({(x^{3} - 2)}^{\frac{1}{5}} + 3)}^{3} - 270 {({(x^{3} - 2)}^{\frac{1}{5}} + 3)}^{2} + 405 ({(x^{3} - 2)}^{\frac{1}{5}} + 3) - 241}$

解题步骤 5.2.4.1.12

将 $81$ 乘以 $5$ 。

$f^{- 1} ({(x^{3} - 2)}^{\frac{1}{5}} + 3) = \sqrt[3]{x^{3} - 2 + 15 {(x^{3} - 2)}^{\frac{4}{5}} + 90 {(x^{3} - 2)}^{\frac{3}{5}} + 270 {(x^{3} - 2)}^{\frac{2}{5}} + 405 {(x^{3} - 2)}^{\frac{1}{5}} + 3^{5} - 15 {({(x^{3} - 2)}^{\frac{1}{5}} + 3)}^{4} + 90 {({(x^{3} - 2)}^{\frac{1}{5}} + 3)}^{3} - 270 {({(x^{3} - 2)}^{\frac{1}{5}} + 3)}^{2} + 405 ({(x^{3} - 2)}^{\frac{1}{5}} + 3) - 241}$

解题步骤 5.2.4.1.13

对 $3$ 进行 $5$ 次方运算。

$f^{- 1} ({(x^{3} - 2)}^{\frac{1}{5}} + 3) = \sqrt[3]{x^{3} - 2 + 15 {(x^{3} - 2)}^{\frac{4}{5}} + 90 {(x^{3} - 2)}^{\frac{3}{5}} + 270 {(x^{3} - 2)}^{\frac{2}{5}} + 405 {(x^{3} - 2)}^{\frac{1}{5}} + 243 - 15 {({(x^{3} - 2)}^{\frac{1}{5}} + 3)}^{4} + 90 {({(x^{3} - 2)}^{\frac{1}{5}} + 3)}^{3} - 270 {({(x^{3} - 2)}^{\frac{1}{5}} + 3)}^{2} + 405 ({(x^{3} - 2)}^{\frac{1}{5}} + 3) - 241}$

解题步骤 5.2.4.2

将 $- 2$ 和 $243$ 相加。

解题步骤 5.2.5

使用二项式定理。

$f^{- 1} ({(x^{3} - 2)}^{\frac{1}{5}} + 3) = \sqrt[3]{x^{3} + 15 {(x^{3} - 2)}^{\frac{4}{5}} + 90 {(x^{3} - 2)}^{\frac{3}{5}} + 270 {(x^{3} - 2)}^{\frac{2}{5}} + 405 {(x^{3} - 2)}^{\frac{1}{5}} + 241 - 15 ({({(x^{3} - 2)}^{\frac{1}{5}})}^{4} + 4 {({(x^{3} - 2)}^{\frac{1}{5}})}^{3} \cdot 3 + 6 {({(x^{3} - 2)}^{\frac{1}{5}})}^{2} \cdot 3^{2} + 4 {(x^{3} - 2)}^{\frac{1}{5}} \cdot 3^{3} + 3^{4}) + 90 {({(x^{3} - 2)}^{\frac{1}{5}} + 3)}^{3} - 270 {({(x^{3} - 2)}^{\frac{1}{5}} + 3)}^{2} + 405 ({(x^{3} - 2)}^{\frac{1}{5}} + 3) - 241}$

解题步骤 5.2.6

化简项。

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解题步骤 5.2.6.1

化简每一项。

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解题步骤 5.2.6.1.1

将 ${({(x^{3} - 2)}^{\frac{1}{5}})}^{4}$ 中的指数相乘。

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解题步骤 5.2.6.1.1.1

运用幂法则并将指数相乘， ${(a^{m})}^{n} = a^{m n}$ 。

$f^{- 1} ({(x^{3} - 2)}^{\frac{1}{5}} + 3) = \sqrt[3]{x^{3} + 15 {(x^{3} - 2)}^{\frac{4}{5}} + 90 {(x^{3} - 2)}^{\frac{3}{5}} + 270 {(x^{3} - 2)}^{\frac{2}{5}} + 405 {(x^{3} - 2)}^{\frac{1}{5}} + 241 - 15 ({(x^{3} - 2)}^{\frac{1}{5} \cdot 4} + 4 {({(x^{3} - 2)}^{\frac{1}{5}})}^{3} \cdot 3 + 6 {({(x^{3} - 2)}^{\frac{1}{5}})}^{2} \cdot 3^{2} + 4 {(x^{3} - 2)}^{\frac{1}{5}} \cdot 3^{3} + 3^{4}) + 90 {({(x^{3} - 2)}^{\frac{1}{5}} + 3)}^{3} - 270 {({(x^{3} - 2)}^{\frac{1}{5}} + 3)}^{2} + 405 ({(x^{3} - 2)}^{\frac{1}{5}} + 3) - 241}$

解题步骤 5.2.6.1.1.2

组合 $\frac{1}{5}$ 和 $4$ 。

$f^{- 1} ({(x^{3} - 2)}^{\frac{1}{5}} + 3) = \sqrt[3]{x^{3} + 15 {(x^{3} - 2)}^{\frac{4}{5}} + 90 {(x^{3} - 2)}^{\frac{3}{5}} + 270 {(x^{3} - 2)}^{\frac{2}{5}} + 405 {(x^{3} - 2)}^{\frac{1}{5}} + 241 - 15 ({(x^{3} - 2)}^{\frac{4}{5}} + 4 {({(x^{3} - 2)}^{\frac{1}{5}})}^{3} \cdot 3 + 6 {({(x^{3} - 2)}^{\frac{1}{5}})}^{2} \cdot 3^{2} + 4 {(x^{3} - 2)}^{\frac{1}{5}} \cdot 3^{3} + 3^{4}) + 90 {({(x^{3} - 2)}^{\frac{1}{5}} + 3)}^{3} - 270 {({(x^{3} - 2)}^{\frac{1}{5}} + 3)}^{2} + 405 ({(x^{3} - 2)}^{\frac{1}{5}} + 3) - 241}$

解题步骤 5.2.6.1.2

将 ${({(x^{3} - 2)}^{\frac{1}{5}})}^{3}$ 中的指数相乘。

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解题步骤 5.2.6.1.2.1

运用幂法则并将指数相乘， ${(a^{m})}^{n} = a^{m n}$ 。

$f^{- 1} ({(x^{3} - 2)}^{\frac{1}{5}} + 3) = \sqrt[3]{x^{3} + 15 {(x^{3} - 2)}^{\frac{4}{5}} + 90 {(x^{3} - 2)}^{\frac{3}{5}} + 270 {(x^{3} - 2)}^{\frac{2}{5}} + 405 {(x^{3} - 2)}^{\frac{1}{5}} + 241 - 15 ({(x^{3} - 2)}^{\frac{4}{5}} + 4 {(x^{3} - 2)}^{\frac{1}{5} \cdot 3} \cdot 3 + 6 {({(x^{3} - 2)}^{\frac{1}{5}})}^{2} \cdot 3^{2} + 4 {(x^{3} - 2)}^{\frac{1}{5}} \cdot 3^{3} + 3^{4}) + 90 {({(x^{3} - 2)}^{\frac{1}{5}} + 3)}^{3} - 270 {({(x^{3} - 2)}^{\frac{1}{5}} + 3)}^{2} + 405 ({(x^{3} - 2)}^{\frac{1}{5}} + 3) - 241}$

解题步骤 5.2.6.1.2.2

组合 $\frac{1}{5}$ 和 $3$ 。

$f^{- 1} ({(x^{3} - 2)}^{\frac{1}{5}} + 3) = \sqrt[3]{x^{3} + 15 {(x^{3} - 2)}^{\frac{4}{5}} + 90 {(x^{3} - 2)}^{\frac{3}{5}} + 270 {(x^{3} - 2)}^{\frac{2}{5}} + 405 {(x^{3} - 2)}^{\frac{1}{5}} + 241 - 15 ({(x^{3} - 2)}^{\frac{4}{5}} + 4 {(x^{3} - 2)}^{\frac{3}{5}} \cdot 3 + 6 {({(x^{3} - 2)}^{\frac{1}{5}})}^{2} \cdot 3^{2} + 4 {(x^{3} - 2)}^{\frac{1}{5}} \cdot 3^{3} + 3^{4}) + 90 {({(x^{3} - 2)}^{\frac{1}{5}} + 3)}^{3} - 270 {({(x^{3} - 2)}^{\frac{1}{5}} + 3)}^{2} + 405 ({(x^{3} - 2)}^{\frac{1}{5}} + 3) - 241}$

解题步骤 5.2.6.1.3

将 $3$ 乘以 $4$ 。

$f^{- 1} ({(x^{3} - 2)}^{\frac{1}{5}} + 3) = \sqrt[3]{x^{3} + 15 {(x^{3} - 2)}^{\frac{4}{5}} + 90 {(x^{3} - 2)}^{\frac{3}{5}} + 270 {(x^{3} - 2)}^{\frac{2}{5}} + 405 {(x^{3} - 2)}^{\frac{1}{5}} + 241 - 15 ({(x^{3} - 2)}^{\frac{4}{5}} + 12 {(x^{3} - 2)}^{\frac{3}{5}} + 6 {({(x^{3} - 2)}^{\frac{1}{5}})}^{2} \cdot 3^{2} + 4 {(x^{3} - 2)}^{\frac{1}{5}} \cdot 3^{3} + 3^{4}) + 90 {({(x^{3} - 2)}^{\frac{1}{5}} + 3)}^{3} - 270 {({(x^{3} - 2)}^{\frac{1}{5}} + 3)}^{2} + 405 ({(x^{3} - 2)}^{\frac{1}{5}} + 3) - 241}$

解题步骤 5.2.6.1.4

将 ${({(x^{3} - 2)}^{\frac{1}{5}})}^{2}$ 中的指数相乘。

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解题步骤 5.2.6.1.4.1

运用幂法则并将指数相乘， ${(a^{m})}^{n} = a^{m n}$ 。

$f^{- 1} ({(x^{3} - 2)}^{\frac{1}{5}} + 3) = \sqrt[3]{x^{3} + 15 {(x^{3} - 2)}^{\frac{4}{5}} + 90 {(x^{3} - 2)}^{\frac{3}{5}} + 270 {(x^{3} - 2)}^{\frac{2}{5}} + 405 {(x^{3} - 2)}^{\frac{1}{5}} + 241 - 15 ({(x^{3} - 2)}^{\frac{4}{5}} + 12 {(x^{3} - 2)}^{\frac{3}{5}} + 6 {(x^{3} - 2)}^{\frac{1}{5} \cdot 2} \cdot 3^{2} + 4 {(x^{3} - 2)}^{\frac{1}{5}} \cdot 3^{3} + 3^{4}) + 90 {({(x^{3} - 2)}^{\frac{1}{5}} + 3)}^{3} - 270 {({(x^{3} - 2)}^{\frac{1}{5}} + 3)}^{2} + 405 ({(x^{3} - 2)}^{\frac{1}{5}} + 3) - 241}$

解题步骤 5.2.6.1.4.2

组合 $\frac{1}{5}$ 和 $2$ 。

$f^{- 1} ({(x^{3} - 2)}^{\frac{1}{5}} + 3) = \sqrt[3]{x^{3} + 15 {(x^{3} - 2)}^{\frac{4}{5}} + 90 {(x^{3} - 2)}^{\frac{3}{5}} + 270 {(x^{3} - 2)}^{\frac{2}{5}} + 405 {(x^{3} - 2)}^{\frac{1}{5}} + 241 - 15 ({(x^{3} - 2)}^{\frac{4}{5}} + 12 {(x^{3} - 2)}^{\frac{3}{5}} + 6 {(x^{3} - 2)}^{\frac{2}{5}} \cdot 3^{2} + 4 {(x^{3} - 2)}^{\frac{1}{5}} \cdot 3^{3} + 3^{4}) + 90 {({(x^{3} - 2)}^{\frac{1}{5}} + 3)}^{3} - 270 {({(x^{3} - 2)}^{\frac{1}{5}} + 3)}^{2} + 405 ({(x^{3} - 2)}^{\frac{1}{5}} + 3) - 241}$

解题步骤 5.2.6.1.5

对 $3$ 进行 $2$ 次方运算。

$f^{- 1} ({(x^{3} - 2)}^{\frac{1}{5}} + 3) = \sqrt[3]{x^{3} + 15 {(x^{3} - 2)}^{\frac{4}{5}} + 90 {(x^{3} - 2)}^{\frac{3}{5}} + 270 {(x^{3} - 2)}^{\frac{2}{5}} + 405 {(x^{3} - 2)}^{\frac{1}{5}} + 241 - 15 ({(x^{3} - 2)}^{\frac{4}{5}} + 12 {(x^{3} - 2)}^{\frac{3}{5}} + 6 {(x^{3} - 2)}^{\frac{2}{5}} \cdot 9 + 4 {(x^{3} - 2)}^{\frac{1}{5}} \cdot 3^{3} + 3^{4}) + 90 {({(x^{3} - 2)}^{\frac{1}{5}} + 3)}^{3} - 270 {({(x^{3} - 2)}^{\frac{1}{5}} + 3)}^{2} + 405 ({(x^{3} - 2)}^{\frac{1}{5}} + 3) - 241}$

解题步骤 5.2.6.1.6

将 $9$ 乘以 $6$ 。

$f^{- 1} ({(x^{3} - 2)}^{\frac{1}{5}} + 3) = \sqrt[3]{x^{3} + 15 {(x^{3} - 2)}^{\frac{4}{5}} + 90 {(x^{3} - 2)}^{\frac{3}{5}} + 270 {(x^{3} - 2)}^{\frac{2}{5}} + 405 {(x^{3} - 2)}^{\frac{1}{5}} + 241 - 15 ({(x^{3} - 2)}^{\frac{4}{5}} + 12 {(x^{3} - 2)}^{\frac{3}{5}} + 54 {(x^{3} - 2)}^{\frac{2}{5}} + 4 {(x^{3} - 2)}^{\frac{1}{5}} \cdot 3^{3} + 3^{4}) + 90 {({(x^{3} - 2)}^{\frac{1}{5}} + 3)}^{3} - 270 {({(x^{3} - 2)}^{\frac{1}{5}} + 3)}^{2} + 405 ({(x^{3} - 2)}^{\frac{1}{5}} + 3) - 241}$

解题步骤 5.2.6.1.7

对 $3$ 进行 $3$ 次方运算。

$f^{- 1} ({(x^{3} - 2)}^{\frac{1}{5}} + 3) = \sqrt[3]{x^{3} + 15 {(x^{3} - 2)}^{\frac{4}{5}} + 90 {(x^{3} - 2)}^{\frac{3}{5}} + 270 {(x^{3} - 2)}^{\frac{2}{5}} + 405 {(x^{3} - 2)}^{\frac{1}{5}} + 241 - 15 ({(x^{3} - 2)}^{\frac{4}{5}} + 12 {(x^{3} - 2)}^{\frac{3}{5}} + 54 {(x^{3} - 2)}^{\frac{2}{5}} + 4 {(x^{3} - 2)}^{\frac{1}{5}} \cdot 27 + 3^{4}) + 90 {({(x^{3} - 2)}^{\frac{1}{5}} + 3)}^{3} - 270 {({(x^{3} - 2)}^{\frac{1}{5}} + 3)}^{2} + 405 ({(x^{3} - 2)}^{\frac{1}{5}} + 3) - 241}$

解题步骤 5.2.6.1.8

将 $27$ 乘以 $4$ 。

$f^{- 1} ({(x^{3} - 2)}^{\frac{1}{5}} + 3) = \sqrt[3]{x^{3} + 15 {(x^{3} - 2)}^{\frac{4}{5}} + 90 {(x^{3} - 2)}^{\frac{3}{5}} + 270 {(x^{3} - 2)}^{\frac{2}{5}} + 405 {(x^{3} - 2)}^{\frac{1}{5}} + 241 - 15 ({(x^{3} - 2)}^{\frac{4}{5}} + 12 {(x^{3} - 2)}^{\frac{3}{5}} + 54 {(x^{3} - 2)}^{\frac{2}{5}} + 108 {(x^{3} - 2)}^{\frac{1}{5}} + 3^{4}) + 90 {({(x^{3} - 2)}^{\frac{1}{5}} + 3)}^{3} - 270 {({(x^{3} - 2)}^{\frac{1}{5}} + 3)}^{2} + 405 ({(x^{3} - 2)}^{\frac{1}{5}} + 3) - 241}$

解题步骤 5.2.6.1.9

对 $3$ 进行 $4$ 次方运算。

$f^{- 1} ({(x^{3} - 2)}^{\frac{1}{5}} + 3) = \sqrt[3]{x^{3} + 15 {(x^{3} - 2)}^{\frac{4}{5}} + 90 {(x^{3} - 2)}^{\frac{3}{5}} + 270 {(x^{3} - 2)}^{\frac{2}{5}} + 405 {(x^{3} - 2)}^{\frac{1}{5}} + 241 - 15 ({(x^{3} - 2)}^{\frac{4}{5}} + 12 {(x^{3} - 2)}^{\frac{3}{5}} + 54 {(x^{3} - 2)}^{\frac{2}{5}} + 108 {(x^{3} - 2)}^{\frac{1}{5}} + 81) + 90 {({(x^{3} - 2)}^{\frac{1}{5}} + 3)}^{3} - 270 {({(x^{3} - 2)}^{\frac{1}{5}} + 3)}^{2} + 405 ({(x^{3} - 2)}^{\frac{1}{5}} + 3) - 241}$

解题步骤 5.2.6.2

运用分配律。

$f^{- 1} ({(x^{3} - 2)}^{\frac{1}{5}} + 3) = \sqrt[3]{x^{3} + 15 {(x^{3} - 2)}^{\frac{4}{5}} + 90 {(x^{3} - 2)}^{\frac{3}{5}} + 270 {(x^{3} - 2)}^{\frac{2}{5}} + 405 {(x^{3} - 2)}^{\frac{1}{5}} + 241 - 15 {(x^{3} - 2)}^{\frac{4}{5}} - 15 (12 {(x^{3} - 2)}^{\frac{3}{5}}) - 15 (54 {(x^{3} - 2)}^{\frac{2}{5}}) - 15 (108 {(x^{3} - 2)}^{\frac{1}{5}}) - 15 \cdot 81 + 90 {({(x^{3} - 2)}^{\frac{1}{5}} + 3)}^{3} - 270 {({(x^{3} - 2)}^{\frac{1}{5}} + 3)}^{2} + 405 ({(x^{3} - 2)}^{\frac{1}{5}} + 3) - 241}$

解题步骤 5.2.7

化简。

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解题步骤 5.2.7.1

将 $12$ 乘以 $- 15$ 。

$f^{- 1} ({(x^{3} - 2)}^{\frac{1}{5}} + 3) = \sqrt[3]{x^{3} + 15 {(x^{3} - 2)}^{\frac{4}{5}} + 90 {(x^{3} - 2)}^{\frac{3}{5}} + 270 {(x^{3} - 2)}^{\frac{2}{5}} + 405 {(x^{3} - 2)}^{\frac{1}{5}} + 241 - 15 {(x^{3} - 2)}^{\frac{4}{5}} - 180 {(x^{3} - 2)}^{\frac{3}{5}} - 15 (54 {(x^{3} - 2)}^{\frac{2}{5}}) - 15 (108 {(x^{3} - 2)}^{\frac{1}{5}}) - 15 \cdot 81 + 90 {({(x^{3} - 2)}^{\frac{1}{5}} + 3)}^{3} - 270 {({(x^{3} - 2)}^{\frac{1}{5}} + 3)}^{2} + 405 ({(x^{3} - 2)}^{\frac{1}{5}} + 3) - 241}$

解题步骤 5.2.7.2

将 $54$ 乘以 $- 15$ 。

$f^{- 1} ({(x^{3} - 2)}^{\frac{1}{5}} + 3) = \sqrt[3]{x^{3} + 15 {(x^{3} - 2)}^{\frac{4}{5}} + 90 {(x^{3} - 2)}^{\frac{3}{5}} + 270 {(x^{3} - 2)}^{\frac{2}{5}} + 405 {(x^{3} - 2)}^{\frac{1}{5}} + 241 - 15 {(x^{3} - 2)}^{\frac{4}{5}} - 180 {(x^{3} - 2)}^{\frac{3}{5}} - 810 {(x^{3} - 2)}^{\frac{2}{5}} - 15 (108 {(x^{3} - 2)}^{\frac{1}{5}}) - 15 \cdot 81 + 90 {({(x^{3} - 2)}^{\frac{1}{5}} + 3)}^{3} - 270 {({(x^{3} - 2)}^{\frac{1}{5}} + 3)}^{2} + 405 ({(x^{3} - 2)}^{\frac{1}{5}} + 3) - 241}$

解题步骤 5.2.7.3

将 $108$ 乘以 $- 15$ 。

$f^{- 1} ({(x^{3} - 2)}^{\frac{1}{5}} + 3) = \sqrt[3]{x^{3} + 15 {(x^{3} - 2)}^{\frac{4}{5}} + 90 {(x^{3} - 2)}^{\frac{3}{5}} + 270 {(x^{3} - 2)}^{\frac{2}{5}} + 405 {(x^{3} - 2)}^{\frac{1}{5}} + 241 - 15 {(x^{3} - 2)}^{\frac{4}{5}} - 180 {(x^{3} - 2)}^{\frac{3}{5}} - 810 {(x^{3} - 2)}^{\frac{2}{5}} - 1620 {(x^{3} - 2)}^{\frac{1}{5}} - 15 \cdot 81 + 90 {({(x^{3} - 2)}^{\frac{1}{5}} + 3)}^{3} - 270 {({(x^{3} - 2)}^{\frac{1}{5}} + 3)}^{2} + 405 ({(x^{3} - 2)}^{\frac{1}{5}} + 3) - 241}$

解题步骤 5.2.7.4

将 $- 15$ 乘以 $81$ 。

$f^{- 1} ({(x^{3} - 2)}^{\frac{1}{5}} + 3) = \sqrt[3]{x^{3} + 15 {(x^{3} - 2)}^{\frac{4}{5}} + 90 {(x^{3} - 2)}^{\frac{3}{5}} + 270 {(x^{3} - 2)}^{\frac{2}{5}} + 405 {(x^{3} - 2)}^{\frac{1}{5}} + 241 - 15 {(x^{3} - 2)}^{\frac{4}{5}} - 180 {(x^{3} - 2)}^{\frac{3}{5}} - 810 {(x^{3} - 2)}^{\frac{2}{5}} - 1620 {(x^{3} - 2)}^{\frac{1}{5}} - 1215 + 90 {({(x^{3} - 2)}^{\frac{1}{5}} + 3)}^{3} - 270 {({(x^{3} - 2)}^{\frac{1}{5}} + 3)}^{2} + 405 ({(x^{3} - 2)}^{\frac{1}{5}} + 3) - 241}$

解题步骤 5.2.8

使用二项式定理。

$f^{- 1} ({(x^{3} - 2)}^{\frac{1}{5}} + 3) = \sqrt[3]{x^{3} + 15 {(x^{3} - 2)}^{\frac{4}{5}} + 90 {(x^{3} - 2)}^{\frac{3}{5}} + 270 {(x^{3} - 2)}^{\frac{2}{5}} + 405 {(x^{3} - 2)}^{\frac{1}{5}} + 241 - 15 {(x^{3} - 2)}^{\frac{4}{5}} - 180 {(x^{3} - 2)}^{\frac{3}{5}} - 810 {(x^{3} - 2)}^{\frac{2}{5}} - 1620 {(x^{3} - 2)}^{\frac{1}{5}} - 1215 + 90 ({({(x^{3} - 2)}^{\frac{1}{5}})}^{3} + 3 {({(x^{3} - 2)}^{\frac{1}{5}})}^{2} \cdot 3 + 3 {(x^{3} - 2)}^{\frac{1}{5}} \cdot 3^{2} + 3^{3}) - 270 {({(x^{3} - 2)}^{\frac{1}{5}} + 3)}^{2} + 405 ({(x^{3} - 2)}^{\frac{1}{5}} + 3) - 241}$

解题步骤 5.2.9

化简项。

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解题步骤 5.2.9.1

化简每一项。

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解题步骤 5.2.9.1.1

将 ${({(x^{3} - 2)}^{\frac{1}{5}})}^{3}$ 中的指数相乘。

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解题步骤 5.2.9.1.1.1

运用幂法则并将指数相乘， ${(a^{m})}^{n} = a^{m n}$ 。

$f^{- 1} ({(x^{3} - 2)}^{\frac{1}{5}} + 3) = \sqrt[3]{x^{3} + 15 {(x^{3} - 2)}^{\frac{4}{5}} + 90 {(x^{3} - 2)}^{\frac{3}{5}} + 270 {(x^{3} - 2)}^{\frac{2}{5}} + 405 {(x^{3} - 2)}^{\frac{1}{5}} + 241 - 15 {(x^{3} - 2)}^{\frac{4}{5}} - 180 {(x^{3} - 2)}^{\frac{3}{5}} - 810 {(x^{3} - 2)}^{\frac{2}{5}} - 1620 {(x^{3} - 2)}^{\frac{1}{5}} - 1215 + 90 ({(x^{3} - 2)}^{\frac{1}{5} \cdot 3} + 3 {({(x^{3} - 2)}^{\frac{1}{5}})}^{2} \cdot 3 + 3 {(x^{3} - 2)}^{\frac{1}{5}} \cdot 3^{2} + 3^{3}) - 270 {({(x^{3} - 2)}^{\frac{1}{5}} + 3)}^{2} + 405 ({(x^{3} - 2)}^{\frac{1}{5}} + 3) - 241}$

解题步骤 5.2.9.1.1.2

组合 $\frac{1}{5}$ 和 $3$ 。

$f^{- 1} ({(x^{3} - 2)}^{\frac{1}{5}} + 3) = \sqrt[3]{x^{3} + 15 {(x^{3} - 2)}^{\frac{4}{5}} + 90 {(x^{3} - 2)}^{\frac{3}{5}} + 270 {(x^{3} - 2)}^{\frac{2}{5}} + 405 {(x^{3} - 2)}^{\frac{1}{5}} + 241 - 15 {(x^{3} - 2)}^{\frac{4}{5}} - 180 {(x^{3} - 2)}^{\frac{3}{5}} - 810 {(x^{3} - 2)}^{\frac{2}{5}} - 1620 {(x^{3} - 2)}^{\frac{1}{5}} - 1215 + 90 ({(x^{3} - 2)}^{\frac{3}{5}} + 3 {({(x^{3} - 2)}^{\frac{1}{5}})}^{2} \cdot 3 + 3 {(x^{3} - 2)}^{\frac{1}{5}} \cdot 3^{2} + 3^{3}) - 270 {({(x^{3} - 2)}^{\frac{1}{5}} + 3)}^{2} + 405 ({(x^{3} - 2)}^{\frac{1}{5}} + 3) - 241}$

解题步骤 5.2.9.1.2

将 ${({(x^{3} - 2)}^{\frac{1}{5}})}^{2}$ 中的指数相乘。

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解题步骤 5.2.9.1.2.1

运用幂法则并将指数相乘， ${(a^{m})}^{n} = a^{m n}$ 。

$f^{- 1} ({(x^{3} - 2)}^{\frac{1}{5}} + 3) = \sqrt[3]{x^{3} + 15 {(x^{3} - 2)}^{\frac{4}{5}} + 90 {(x^{3} - 2)}^{\frac{3}{5}} + 270 {(x^{3} - 2)}^{\frac{2}{5}} + 405 {(x^{3} - 2)}^{\frac{1}{5}} + 241 - 15 {(x^{3} - 2)}^{\frac{4}{5}} - 180 {(x^{3} - 2)}^{\frac{3}{5}} - 810 {(x^{3} - 2)}^{\frac{2}{5}} - 1620 {(x^{3} - 2)}^{\frac{1}{5}} - 1215 + 90 ({(x^{3} - 2)}^{\frac{3}{5}} + 3 {(x^{3} - 2)}^{\frac{1}{5} \cdot 2} \cdot 3 + 3 {(x^{3} - 2)}^{\frac{1}{5}} \cdot 3^{2} + 3^{3}) - 270 {({(x^{3} - 2)}^{\frac{1}{5}} + 3)}^{2} + 405 ({(x^{3} - 2)}^{\frac{1}{5}} + 3) - 241}$

解题步骤 5.2.9.1.2.2

组合 $\frac{1}{5}$ 和 $2$ 。

$f^{- 1} ({(x^{3} - 2)}^{\frac{1}{5}} + 3) = \sqrt[3]{x^{3} + 15 {(x^{3} - 2)}^{\frac{4}{5}} + 90 {(x^{3} - 2)}^{\frac{3}{5}} + 270 {(x^{3} - 2)}^{\frac{2}{5}} + 405 {(x^{3} - 2)}^{\frac{1}{5}} + 241 - 15 {(x^{3} - 2)}^{\frac{4}{5}} - 180 {(x^{3} - 2)}^{\frac{3}{5}} - 810 {(x^{3} - 2)}^{\frac{2}{5}} - 1620 {(x^{3} - 2)}^{\frac{1}{5}} - 1215 + 90 ({(x^{3} - 2)}^{\frac{3}{5}} + 3 {(x^{3} - 2)}^{\frac{2}{5}} \cdot 3 + 3 {(x^{3} - 2)}^{\frac{1}{5}} \cdot 3^{2} + 3^{3}) - 270 {({(x^{3} - 2)}^{\frac{1}{5}} + 3)}^{2} + 405 ({(x^{3} - 2)}^{\frac{1}{5}} + 3) - 241}$

解题步骤 5.2.9.1.3

将 $3$ 乘以 $3$ 。

$f^{- 1} ({(x^{3} - 2)}^{\frac{1}{5}} + 3) = \sqrt[3]{x^{3} + 15 {(x^{3} - 2)}^{\frac{4}{5}} + 90 {(x^{3} - 2)}^{\frac{3}{5}} + 270 {(x^{3} - 2)}^{\frac{2}{5}} + 405 {(x^{3} - 2)}^{\frac{1}{5}} + 241 - 15 {(x^{3} - 2)}^{\frac{4}{5}} - 180 {(x^{3} - 2)}^{\frac{3}{5}} - 810 {(x^{3} - 2)}^{\frac{2}{5}} - 1620 {(x^{3} - 2)}^{\frac{1}{5}} - 1215 + 90 ({(x^{3} - 2)}^{\frac{3}{5}} + 9 {(x^{3} - 2)}^{\frac{2}{5}} + 3 {(x^{3} - 2)}^{\frac{1}{5}} \cdot 3^{2} + 3^{3}) - 270 {({(x^{3} - 2)}^{\frac{1}{5}} + 3)}^{2} + 405 ({(x^{3} - 2)}^{\frac{1}{5}} + 3) - 241}$

解题步骤 5.2.9.1.4

通过指数相加将 $3$ 乘以 $3^{2}$ 。

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解题步骤 5.2.9.1.4.1

移动 $3^{2}$ 。

$f^{- 1} ({(x^{3} - 2)}^{\frac{1}{5}} + 3) = \sqrt[3]{x^{3} + 15 {(x^{3} - 2)}^{\frac{4}{5}} + 90 {(x^{3} - 2)}^{\frac{3}{5}} + 270 {(x^{3} - 2)}^{\frac{2}{5}} + 405 {(x^{3} - 2)}^{\frac{1}{5}} + 241 - 15 {(x^{3} - 2)}^{\frac{4}{5}} - 180 {(x^{3} - 2)}^{\frac{3}{5}} - 810 {(x^{3} - 2)}^{\frac{2}{5}} - 1620 {(x^{3} - 2)}^{\frac{1}{5}} - 1215 + 90 ({(x^{3} - 2)}^{\frac{3}{5}} + 9 {(x^{3} - 2)}^{\frac{2}{5}} + 3^{2} \cdot (3 {(x^{3} - 2)}^{\frac{1}{5}}) + 3^{3}) - 270 {({(x^{3} - 2)}^{\frac{1}{5}} + 3)}^{2} + 405 ({(x^{3} - 2)}^{\frac{1}{5}} + 3) - 241}$

解题步骤 5.2.9.1.4.2

将 $3^{2}$ 乘以 $3$ 。

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解题步骤 5.2.9.1.4.2.1

对 $3$ 进行 $1$ 次方运算。

$f^{- 1} ({(x^{3} - 2)}^{\frac{1}{5}} + 3) = \sqrt[3]{x^{3} + 15 {(x^{3} - 2)}^{\frac{4}{5}} + 90 {(x^{3} - 2)}^{\frac{3}{5}} + 270 {(x^{3} - 2)}^{\frac{2}{5}} + 405 {(x^{3} - 2)}^{\frac{1}{5}} + 241 - 15 {(x^{3} - 2)}^{\frac{4}{5}} - 180 {(x^{3} - 2)}^{\frac{3}{5}} - 810 {(x^{3} - 2)}^{\frac{2}{5}} - 1620 {(x^{3} - 2)}^{\frac{1}{5}} - 1215 + 90 ({(x^{3} - 2)}^{\frac{3}{5}} + 9 {(x^{3} - 2)}^{\frac{2}{5}} + 3^{2} \cdot (3 {(x^{3} - 2)}^{\frac{1}{5}}) + 3^{3}) - 270 {({(x^{3} - 2)}^{\frac{1}{5}} + 3)}^{2} + 405 ({(x^{3} - 2)}^{\frac{1}{5}} + 3) - 241}$

解题步骤 5.2.9.1.4.2.2

使用幂法则 $a^{m} a^{n} = a^{m + n}$ 合并指数。

$f^{- 1} ({(x^{3} - 2)}^{\frac{1}{5}} + 3) = \sqrt[3]{x^{3} + 15 {(x^{3} - 2)}^{\frac{4}{5}} + 90 {(x^{3} - 2)}^{\frac{3}{5}} + 270 {(x^{3} - 2)}^{\frac{2}{5}} + 405 {(x^{3} - 2)}^{\frac{1}{5}} + 241 - 15 {(x^{3} - 2)}^{\frac{4}{5}} - 180 {(x^{3} - 2)}^{\frac{3}{5}} - 810 {(x^{3} - 2)}^{\frac{2}{5}} - 1620 {(x^{3} - 2)}^{\frac{1}{5}} - 1215 + 90 ({(x^{3} - 2)}^{\frac{3}{5}} + 9 {(x^{3} - 2)}^{\frac{2}{5}} + 3^{2 + 1} {(x^{3} - 2)}^{\frac{1}{5}} + 3^{3}) - 270 {({(x^{3} - 2)}^{\frac{1}{5}} + 3)}^{2} + 405 ({(x^{3} - 2)}^{\frac{1}{5}} + 3) - 241}$

解题步骤 5.2.9.1.4.3

将 $2$ 和 $1$ 相加。

$f^{- 1} ({(x^{3} - 2)}^{\frac{1}{5}} + 3) = \sqrt[3]{x^{3} + 15 {(x^{3} - 2)}^{\frac{4}{5}} + 90 {(x^{3} - 2)}^{\frac{3}{5}} + 270 {(x^{3} - 2)}^{\frac{2}{5}} + 405 {(x^{3} - 2)}^{\frac{1}{5}} + 241 - 15 {(x^{3} - 2)}^{\frac{4}{5}} - 180 {(x^{3} - 2)}^{\frac{3}{5}} - 810 {(x^{3} - 2)}^{\frac{2}{5}} - 1620 {(x^{3} - 2)}^{\frac{1}{5}} - 1215 + 90 ({(x^{3} - 2)}^{\frac{3}{5}} + 9 {(x^{3} - 2)}^{\frac{2}{5}} + 3^{3} {(x^{3} - 2)}^{\frac{1}{5}} + 3^{3}) - 270 {({(x^{3} - 2)}^{\frac{1}{5}} + 3)}^{2} + 405 ({(x^{3} - 2)}^{\frac{1}{5}} + 3) - 241}$

解题步骤 5.2.9.1.5

对 $3$ 进行 $3$ 次方运算。

$f^{- 1} ({(x^{3} - 2)}^{\frac{1}{5}} + 3) = \sqrt[3]{x^{3} + 15 {(x^{3} - 2)}^{\frac{4}{5}} + 90 {(x^{3} - 2)}^{\frac{3}{5}} + 270 {(x^{3} - 2)}^{\frac{2}{5}} + 405 {(x^{3} - 2)}^{\frac{1}{5}} + 241 - 15 {(x^{3} - 2)}^{\frac{4}{5}} - 180 {(x^{3} - 2)}^{\frac{3}{5}} - 810 {(x^{3} - 2)}^{\frac{2}{5}} - 1620 {(x^{3} - 2)}^{\frac{1}{5}} - 1215 + 90 ({(x^{3} - 2)}^{\frac{3}{5}} + 9 {(x^{3} - 2)}^{\frac{2}{5}} + 27 {(x^{3} - 2)}^{\frac{1}{5}} + 3^{3}) - 270 {({(x^{3} - 2)}^{\frac{1}{5}} + 3)}^{2} + 405 ({(x^{3} - 2)}^{\frac{1}{5}} + 3) - 241}$

解题步骤 5.2.9.1.6

对 $3$ 进行 $3$ 次方运算。

$f^{- 1} ({(x^{3} - 2)}^{\frac{1}{5}} + 3) = \sqrt[3]{x^{3} + 15 {(x^{3} - 2)}^{\frac{4}{5}} + 90 {(x^{3} - 2)}^{\frac{3}{5}} + 270 {(x^{3} - 2)}^{\frac{2}{5}} + 405 {(x^{3} - 2)}^{\frac{1}{5}} + 241 - 15 {(x^{3} - 2)}^{\frac{4}{5}} - 180 {(x^{3} - 2)}^{\frac{3}{5}} - 810 {(x^{3} - 2)}^{\frac{2}{5}} - 1620 {(x^{3} - 2)}^{\frac{1}{5}} - 1215 + 90 ({(x^{3} - 2)}^{\frac{3}{5}} + 9 {(x^{3} - 2)}^{\frac{2}{5}} + 27 {(x^{3} - 2)}^{\frac{1}{5}} + 27) - 270 {({(x^{3} - 2)}^{\frac{1}{5}} + 3)}^{2} + 405 ({(x^{3} - 2)}^{\frac{1}{5}} + 3) - 241}$

解题步骤 5.2.9.2

运用分配律。

$f^{- 1} ({(x^{3} - 2)}^{\frac{1}{5}} + 3) = \sqrt[3]{x^{3} + 15 {(x^{3} - 2)}^{\frac{4}{5}} + 90 {(x^{3} - 2)}^{\frac{3}{5}} + 270 {(x^{3} - 2)}^{\frac{2}{5}} + 405 {(x^{3} - 2)}^{\frac{1}{5}} + 241 - 15 {(x^{3} - 2)}^{\frac{4}{5}} - 180 {(x^{3} - 2)}^{\frac{3}{5}} - 810 {(x^{3} - 2)}^{\frac{2}{5}} - 1620 {(x^{3} - 2)}^{\frac{1}{5}} - 1215 + 90 {(x^{3} - 2)}^{\frac{3}{5}} + 90 (9 {(x^{3} - 2)}^{\frac{2}{5}}) + 90 (27 {(x^{3} - 2)}^{\frac{1}{5}}) + 90 \cdot 27 - 270 {({(x^{3} - 2)}^{\frac{1}{5}} + 3)}^{2} + 405 ({(x^{3} - 2)}^{\frac{1}{5}} + 3) - 241}$

解题步骤 5.2.10

化简。

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解题步骤 5.2.10.1

将 $9$ 乘以 $90$ 。

$f^{- 1} ({(x^{3} - 2)}^{\frac{1}{5}} + 3) = \sqrt[3]{x^{3} + 15 {(x^{3} - 2)}^{\frac{4}{5}} + 90 {(x^{3} - 2)}^{\frac{3}{5}} + 270 {(x^{3} - 2)}^{\frac{2}{5}} + 405 {(x^{3} - 2)}^{\frac{1}{5}} + 241 - 15 {(x^{3} - 2)}^{\frac{4}{5}} - 180 {(x^{3} - 2)}^{\frac{3}{5}} - 810 {(x^{3} - 2)}^{\frac{2}{5}} - 1620 {(x^{3} - 2)}^{\frac{1}{5}} - 1215 + 90 {(x^{3} - 2)}^{\frac{3}{5}} + 810 {(x^{3} - 2)}^{\frac{2}{5}} + 90 (27 {(x^{3} - 2)}^{\frac{1}{5}}) + 90 \cdot 27 - 270 {({(x^{3} - 2)}^{\frac{1}{5}} + 3)}^{2} + 405 ({(x^{3} - 2)}^{\frac{1}{5}} + 3) - 241}$

解题步骤 5.2.10.2

将 $27$ 乘以 $90$ 。

$f^{- 1} ({(x^{3} - 2)}^{\frac{1}{5}} + 3) = \sqrt[3]{x^{3} + 15 {(x^{3} - 2)}^{\frac{4}{5}} + 90 {(x^{3} - 2)}^{\frac{3}{5}} + 270 {(x^{3} - 2)}^{\frac{2}{5}} + 405 {(x^{3} - 2)}^{\frac{1}{5}} + 241 - 15 {(x^{3} - 2)}^{\frac{4}{5}} - 180 {(x^{3} - 2)}^{\frac{3}{5}} - 810 {(x^{3} - 2)}^{\frac{2}{5}} - 1620 {(x^{3} - 2)}^{\frac{1}{5}} - 1215 + 90 {(x^{3} - 2)}^{\frac{3}{5}} + 810 {(x^{3} - 2)}^{\frac{2}{5}} + 2430 {(x^{3} - 2)}^{\frac{1}{5}} + 90 \cdot 27 - 270 {({(x^{3} - 2)}^{\frac{1}{5}} + 3)}^{2} + 405 ({(x^{3} - 2)}^{\frac{1}{5}} + 3) - 241}$

解题步骤 5.2.10.3

将 $90$ 乘以 $27$ 。

$f^{- 1} ({(x^{3} - 2)}^{\frac{1}{5}} + 3) = \sqrt[3]{x^{3} + 15 {(x^{3} - 2)}^{\frac{4}{5}} + 90 {(x^{3} - 2)}^{\frac{3}{5}} + 270 {(x^{3} - 2)}^{\frac{2}{5}} + 405 {(x^{3} - 2)}^{\frac{1}{5}} + 241 - 15 {(x^{3} - 2)}^{\frac{4}{5}} - 180 {(x^{3} - 2)}^{\frac{3}{5}} - 810 {(x^{3} - 2)}^{\frac{2}{5}} - 1620 {(x^{3} - 2)}^{\frac{1}{5}} - 1215 + 90 {(x^{3} - 2)}^{\frac{3}{5}} + 810 {(x^{3} - 2)}^{\frac{2}{5}} + 2430 {(x^{3} - 2)}^{\frac{1}{5}} + 2430 - 270 {({(x^{3} - 2)}^{\frac{1}{5}} + 3)}^{2} + 405 ({(x^{3} - 2)}^{\frac{1}{5}} + 3) - 241}$

解题步骤 5.2.11

将 ${({(x^{3} - 2)}^{\frac{1}{5}} + 3)}^{2}$ 重写为 $({(x^{3} - 2)}^{\frac{1}{5}} + 3) ({(x^{3} - 2)}^{\frac{1}{5}} + 3)$ 。

$f^{- 1} ({(x^{3} - 2)}^{\frac{1}{5}} + 3) = \sqrt[3]{x^{3} + 15 {(x^{3} - 2)}^{\frac{4}{5}} + 90 {(x^{3} - 2)}^{\frac{3}{5}} + 270 {(x^{3} - 2)}^{\frac{2}{5}} + 405 {(x^{3} - 2)}^{\frac{1}{5}} + 241 - 15 {(x^{3} - 2)}^{\frac{4}{5}} - 180 {(x^{3} - 2)}^{\frac{3}{5}} - 810 {(x^{3} - 2)}^{\frac{2}{5}} - 1620 {(x^{3} - 2)}^{\frac{1}{5}} - 1215 + 90 {(x^{3} - 2)}^{\frac{3}{5}} + 810 {(x^{3} - 2)}^{\frac{2}{5}} + 2430 {(x^{3} - 2)}^{\frac{1}{5}} + 2430 - 270 (({(x^{3} - 2)}^{\frac{1}{5}} + 3) ({(x^{3} - 2)}^{\frac{1}{5}} + 3)) + 405 ({(x^{3} - 2)}^{\frac{1}{5}} + 3) - 241}$

解题步骤 5.2.12

使用 FOIL 方法展开 $({(x^{3} - 2)}^{\frac{1}{5}} + 3) ({(x^{3} - 2)}^{\frac{1}{5}} + 3)$ 。

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解题步骤 5.2.12.1

运用分配律。

$f^{- 1} ({(x^{3} - 2)}^{\frac{1}{5}} + 3) = \sqrt[3]{x^{3} + 15 {(x^{3} - 2)}^{\frac{4}{5}} + 90 {(x^{3} - 2)}^{\frac{3}{5}} + 270 {(x^{3} - 2)}^{\frac{2}{5}} + 405 {(x^{3} - 2)}^{\frac{1}{5}} + 241 - 15 {(x^{3} - 2)}^{\frac{4}{5}} - 180 {(x^{3} - 2)}^{\frac{3}{5}} - 810 {(x^{3} - 2)}^{\frac{2}{5}} - 1620 {(x^{3} - 2)}^{\frac{1}{5}} - 1215 + 90 {(x^{3} - 2)}^{\frac{3}{5}} + 810 {(x^{3} - 2)}^{\frac{2}{5}} + 2430 {(x^{3} - 2)}^{\frac{1}{5}} + 2430 - 270 ({(x^{3} - 2)}^{\frac{1}{5}} ({(x^{3} - 2)}^{\frac{1}{5}} + 3) + 3 ({(x^{3} - 2)}^{\frac{1}{5}} + 3)) + 405 ({(x^{3} - 2)}^{\frac{1}{5}} + 3) - 241}$

解题步骤 5.2.12.2

运用分配律。

$f^{- 1} ({(x^{3} - 2)}^{\frac{1}{5}} + 3) = \sqrt[3]{x^{3} + 15 {(x^{3} - 2)}^{\frac{4}{5}} + 90 {(x^{3} - 2)}^{\frac{3}{5}} + 270 {(x^{3} - 2)}^{\frac{2}{5}} + 405 {(x^{3} - 2)}^{\frac{1}{5}} + 241 - 15 {(x^{3} - 2)}^{\frac{4}{5}} - 180 {(x^{3} - 2)}^{\frac{3}{5}} - 810 {(x^{3} - 2)}^{\frac{2}{5}} - 1620 {(x^{3} - 2)}^{\frac{1}{5}} - 1215 + 90 {(x^{3} - 2)}^{\frac{3}{5}} + 810 {(x^{3} - 2)}^{\frac{2}{5}} + 2430 {(x^{3} - 2)}^{\frac{1}{5}} + 2430 - 270 ({(x^{3} - 2)}^{\frac{1}{5}} {(x^{3} - 2)}^{\frac{1}{5}} + {(x^{3} - 2)}^{\frac{1}{5}} \cdot 3 + 3 ({(x^{3} - 2)}^{\frac{1}{5}} + 3)) + 405 ({(x^{3} - 2)}^{\frac{1}{5}} + 3) - 241}$

解题步骤 5.2.12.3

运用分配律。

$f^{- 1} ({(x^{3} - 2)}^{\frac{1}{5}} + 3) = \sqrt[3]{x^{3} + 15 {(x^{3} - 2)}^{\frac{4}{5}} + 90 {(x^{3} - 2)}^{\frac{3}{5}} + 270 {(x^{3} - 2)}^{\frac{2}{5}} + 405 {(x^{3} - 2)}^{\frac{1}{5}} + 241 - 15 {(x^{3} - 2)}^{\frac{4}{5}} - 180 {(x^{3} - 2)}^{\frac{3}{5}} - 810 {(x^{3} - 2)}^{\frac{2}{5}} - 1620 {(x^{3} - 2)}^{\frac{1}{5}} - 1215 + 90 {(x^{3} - 2)}^{\frac{3}{5}} + 810 {(x^{3} - 2)}^{\frac{2}{5}} + 2430 {(x^{3} - 2)}^{\frac{1}{5}} + 2430 - 270 ({(x^{3} - 2)}^{\frac{1}{5}} {(x^{3} - 2)}^{\frac{1}{5}} + {(x^{3} - 2)}^{\frac{1}{5}} \cdot 3 + 3 {(x^{3} - 2)}^{\frac{1}{5}} + 3 \cdot 3) + 405 ({(x^{3} - 2)}^{\frac{1}{5}} + 3) - 241}$

解题步骤 5.2.13

化简并合并同类项。

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解题步骤 5.2.13.1

化简每一项。

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解题步骤 5.2.13.1.1

通过指数相加将 ${(x^{3} - 2)}^{\frac{1}{5}}$ 乘以 ${(x^{3} - 2)}^{\frac{1}{5}}$ 。

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解题步骤 5.2.13.1.1.1

使用幂法则 $a^{m} a^{n} = a^{m + n}$ 合并指数。

$f^{- 1} ({(x^{3} - 2)}^{\frac{1}{5}} + 3) = \sqrt[3]{x^{3} + 15 {(x^{3} - 2)}^{\frac{4}{5}} + 90 {(x^{3} - 2)}^{\frac{3}{5}} + 270 {(x^{3} - 2)}^{\frac{2}{5}} + 405 {(x^{3} - 2)}^{\frac{1}{5}} + 241 - 15 {(x^{3} - 2)}^{\frac{4}{5}} - 180 {(x^{3} - 2)}^{\frac{3}{5}} - 810 {(x^{3} - 2)}^{\frac{2}{5}} - 1620 {(x^{3} - 2)}^{\frac{1}{5}} - 1215 + 90 {(x^{3} - 2)}^{\frac{3}{5}} + 810 {(x^{3} - 2)}^{\frac{2}{5}} + 2430 {(x^{3} - 2)}^{\frac{1}{5}} + 2430 - 270 ({(x^{3} - 2)}^{\frac{1}{5} + \frac{1}{5}} + {(x^{3} - 2)}^{\frac{1}{5}} \cdot 3 + 3 {(x^{3} - 2)}^{\frac{1}{5}} + 3 \cdot 3) + 405 ({(x^{3} - 2)}^{\frac{1}{5}} + 3) - 241}$

解题步骤 5.2.13.1.1.2

在公分母上合并分子。

$f^{- 1} ({(x^{3} - 2)}^{\frac{1}{5}} + 3) = \sqrt[3]{x^{3} + 15 {(x^{3} - 2)}^{\frac{4}{5}} + 90 {(x^{3} - 2)}^{\frac{3}{5}} + 270 {(x^{3} - 2)}^{\frac{2}{5}} + 405 {(x^{3} - 2)}^{\frac{1}{5}} + 241 - 15 {(x^{3} - 2)}^{\frac{4}{5}} - 180 {(x^{3} - 2)}^{\frac{3}{5}} - 810 {(x^{3} - 2)}^{\frac{2}{5}} - 1620 {(x^{3} - 2)}^{\frac{1}{5}} - 1215 + 90 {(x^{3} - 2)}^{\frac{3}{5}} + 810 {(x^{3} - 2)}^{\frac{2}{5}} + 2430 {(x^{3} - 2)}^{\frac{1}{5}} + 2430 - 270 ({(x^{3} - 2)}^{\frac{1 + 1}{5}} + {(x^{3} - 2)}^{\frac{1}{5}} \cdot 3 + 3 {(x^{3} - 2)}^{\frac{1}{5}} + 3 \cdot 3) + 405 ({(x^{3} - 2)}^{\frac{1}{5}} + 3) - 241}$

解题步骤 5.2.13.1.1.3

将 $1$ 和 $1$ 相加。

$f^{- 1} ({(x^{3} - 2)}^{\frac{1}{5}} + 3) = \sqrt[3]{x^{3} + 15 {(x^{3} - 2)}^{\frac{4}{5}} + 90 {(x^{3} - 2)}^{\frac{3}{5}} + 270 {(x^{3} - 2)}^{\frac{2}{5}} + 405 {(x^{3} - 2)}^{\frac{1}{5}} + 241 - 15 {(x^{3} - 2)}^{\frac{4}{5}} - 180 {(x^{3} - 2)}^{\frac{3}{5}} - 810 {(x^{3} - 2)}^{\frac{2}{5}} - 1620 {(x^{3} - 2)}^{\frac{1}{5}} - 1215 + 90 {(x^{3} - 2)}^{\frac{3}{5}} + 810 {(x^{3} - 2)}^{\frac{2}{5}} + 2430 {(x^{3} - 2)}^{\frac{1}{5}} + 2430 - 270 ({(x^{3} - 2)}^{\frac{2}{5}} + {(x^{3} - 2)}^{\frac{1}{5}} \cdot 3 + 3 {(x^{3} - 2)}^{\frac{1}{5}} + 3 \cdot 3) + 405 ({(x^{3} - 2)}^{\frac{1}{5}} + 3) - 241}$

解题步骤 5.2.13.1.2

将 $3$ 移到 ${(x^{3} - 2)}^{\frac{1}{5}}$ 的左侧。

$f^{- 1} ({(x^{3} - 2)}^{\frac{1}{5}} + 3) = \sqrt[3]{x^{3} + 15 {(x^{3} - 2)}^{\frac{4}{5}} + 90 {(x^{3} - 2)}^{\frac{3}{5}} + 270 {(x^{3} - 2)}^{\frac{2}{5}} + 405 {(x^{3} - 2)}^{\frac{1}{5}} + 241 - 15 {(x^{3} - 2)}^{\frac{4}{5}} - 180 {(x^{3} - 2)}^{\frac{3}{5}} - 810 {(x^{3} - 2)}^{\frac{2}{5}} - 1620 {(x^{3} - 2)}^{\frac{1}{5}} - 1215 + 90 {(x^{3} - 2)}^{\frac{3}{5}} + 810 {(x^{3} - 2)}^{\frac{2}{5}} + 2430 {(x^{3} - 2)}^{\frac{1}{5}} + 2430 - 270 ({(x^{3} - 2)}^{\frac{2}{5}} + 3 \cdot {(x^{3} - 2)}^{\frac{1}{5}} + 3 {(x^{3} - 2)}^{\frac{1}{5}} + 3 \cdot 3) + 405 ({(x^{3} - 2)}^{\frac{1}{5}} + 3) - 241}$

解题步骤 5.2.13.1.3

将 $3$ 乘以 $3$ 。

$f^{- 1} ({(x^{3} - 2)}^{\frac{1}{5}} + 3) = \sqrt[3]{x^{3} + 15 {(x^{3} - 2)}^{\frac{4}{5}} + 90 {(x^{3} - 2)}^{\frac{3}{5}} + 270 {(x^{3} - 2)}^{\frac{2}{5}} + 405 {(x^{3} - 2)}^{\frac{1}{5}} + 241 - 15 {(x^{3} - 2)}^{\frac{4}{5}} - 180 {(x^{3} - 2)}^{\frac{3}{5}} - 810 {(x^{3} - 2)}^{\frac{2}{5}} - 1620 {(x^{3} - 2)}^{\frac{1}{5}} - 1215 + 90 {(x^{3} - 2)}^{\frac{3}{5}} + 810 {(x^{3} - 2)}^{\frac{2}{5}} + 2430 {(x^{3} - 2)}^{\frac{1}{5}} + 2430 - 270 ({(x^{3} - 2)}^{\frac{2}{5}} + 3 {(x^{3} - 2)}^{\frac{1}{5}} + 3 {(x^{3} - 2)}^{\frac{1}{5}} + 9) + 405 ({(x^{3} - 2)}^{\frac{1}{5}} + 3) - 241}$

解题步骤 5.2.13.2

将 $3 {(x^{3} - 2)}^{\frac{1}{5}}$ 和 $3 {(x^{3} - 2)}^{\frac{1}{5}}$ 相加。

$f^{- 1} ({(x^{3} - 2)}^{\frac{1}{5}} + 3) = \sqrt[3]{x^{3} + 15 {(x^{3} - 2)}^{\frac{4}{5}} + 90 {(x^{3} - 2)}^{\frac{3}{5}} + 270 {(x^{3} - 2)}^{\frac{2}{5}} + 405 {(x^{3} - 2)}^{\frac{1}{5}} + 241 - 15 {(x^{3} - 2)}^{\frac{4}{5}} - 180 {(x^{3} - 2)}^{\frac{3}{5}} - 810 {(x^{3} - 2)}^{\frac{2}{5}} - 1620 {(x^{3} - 2)}^{\frac{1}{5}} - 1215 + 90 {(x^{3} - 2)}^{\frac{3}{5}} + 810 {(x^{3} - 2)}^{\frac{2}{5}} + 2430 {(x^{3} - 2)}^{\frac{1}{5}} + 2430 - 270 ({(x^{3} - 2)}^{\frac{2}{5}} + 6 {(x^{3} - 2)}^{\frac{1}{5}} + 9) + 405 ({(x^{3} - 2)}^{\frac{1}{5}} + 3) - 241}$

解题步骤 5.2.14

运用分配律。

$f^{- 1} ({(x^{3} - 2)}^{\frac{1}{5}} + 3) = \sqrt[3]{x^{3} + 15 {(x^{3} - 2)}^{\frac{4}{5}} + 90 {(x^{3} - 2)}^{\frac{3}{5}} + 270 {(x^{3} - 2)}^{\frac{2}{5}} + 405 {(x^{3} - 2)}^{\frac{1}{5}} + 241 - 15 {(x^{3} - 2)}^{\frac{4}{5}} - 180 {(x^{3} - 2)}^{\frac{3}{5}} - 810 {(x^{3} - 2)}^{\frac{2}{5}} - 1620 {(x^{3} - 2)}^{\frac{1}{5}} - 1215 + 90 {(x^{3} - 2)}^{\frac{3}{5}} + 810 {(x^{3} - 2)}^{\frac{2}{5}} + 2430 {(x^{3} - 2)}^{\frac{1}{5}} + 2430 - 270 {(x^{3} - 2)}^{\frac{2}{5}} - 270 (6 {(x^{3} - 2)}^{\frac{1}{5}}) - 270 \cdot 9 + 405 ({(x^{3} - 2)}^{\frac{1}{5}} + 3) - 241}$

解题步骤 5.2.15

化简。

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解题步骤 5.2.15.1

将 $6$ 乘以 $- 270$ 。

$f^{- 1} ({(x^{3} - 2)}^{\frac{1}{5}} + 3) = \sqrt[3]{x^{3} + 15 {(x^{3} - 2)}^{\frac{4}{5}} + 90 {(x^{3} - 2)}^{\frac{3}{5}} + 270 {(x^{3} - 2)}^{\frac{2}{5}} + 405 {(x^{3} - 2)}^{\frac{1}{5}} + 241 - 15 {(x^{3} - 2)}^{\frac{4}{5}} - 180 {(x^{3} - 2)}^{\frac{3}{5}} - 810 {(x^{3} - 2)}^{\frac{2}{5}} - 1620 {(x^{3} - 2)}^{\frac{1}{5}} - 1215 + 90 {(x^{3} - 2)}^{\frac{3}{5}} + 810 {(x^{3} - 2)}^{\frac{2}{5}} + 2430 {(x^{3} - 2)}^{\frac{1}{5}} + 2430 - 270 {(x^{3} - 2)}^{\frac{2}{5}} - 1620 {(x^{3} - 2)}^{\frac{1}{5}} - 270 \cdot 9 + 405 ({(x^{3} - 2)}^{\frac{1}{5}} + 3) - 241}$

解题步骤 5.2.15.2

将 $- 270$ 乘以 $9$ 。

$f^{- 1} ({(x^{3} - 2)}^{\frac{1}{5}} + 3) = \sqrt[3]{x^{3} + 15 {(x^{3} - 2)}^{\frac{4}{5}} + 90 {(x^{3} - 2)}^{\frac{3}{5}} + 270 {(x^{3} - 2)}^{\frac{2}{5}} + 405 {(x^{3} - 2)}^{\frac{1}{5}} + 241 - 15 {(x^{3} - 2)}^{\frac{4}{5}} - 180 {(x^{3} - 2)}^{\frac{3}{5}} - 810 {(x^{3} - 2)}^{\frac{2}{5}} - 1620 {(x^{3} - 2)}^{\frac{1}{5}} - 1215 + 90 {(x^{3} - 2)}^{\frac{3}{5}} + 810 {(x^{3} - 2)}^{\frac{2}{5}} + 2430 {(x^{3} - 2)}^{\frac{1}{5}} + 2430 - 270 {(x^{3} - 2)}^{\frac{2}{5}} - 1620 {(x^{3} - 2)}^{\frac{1}{5}} - 2430 + 405 ({(x^{3} - 2)}^{\frac{1}{5}} + 3) - 241}$

解题步骤 5.2.16

运用分配律。

$f^{- 1} ({(x^{3} - 2)}^{\frac{1}{5}} + 3) = \sqrt[3]{x^{3} + 15 {(x^{3} - 2)}^{\frac{4}{5}} + 90 {(x^{3} - 2)}^{\frac{3}{5}} + 270 {(x^{3} - 2)}^{\frac{2}{5}} + 405 {(x^{3} - 2)}^{\frac{1}{5}} + 241 - 15 {(x^{3} - 2)}^{\frac{4}{5}} - 180 {(x^{3} - 2)}^{\frac{3}{5}} - 810 {(x^{3} - 2)}^{\frac{2}{5}} - 1620 {(x^{3} - 2)}^{\frac{1}{5}} - 1215 + 90 {(x^{3} - 2)}^{\frac{3}{5}} + 810 {(x^{3} - 2)}^{\frac{2}{5}} + 2430 {(x^{3} - 2)}^{\frac{1}{5}} + 2430 - 270 {(x^{3} - 2)}^{\frac{2}{5}} - 1620 {(x^{3} - 2)}^{\frac{1}{5}} - 2430 + 405 {(x^{3} - 2)}^{\frac{1}{5}} + 405 \cdot 3 - 241}$

解题步骤 5.2.17

将 $405$ 乘以 $3$ 。

$f^{- 1} ({(x^{3} - 2)}^{\frac{1}{5}} + 3) = \sqrt[3]{x^{3} + 15 {(x^{3} - 2)}^{\frac{4}{5}} + 90 {(x^{3} - 2)}^{\frac{3}{5}} + 270 {(x^{3} - 2)}^{\frac{2}{5}} + 405 {(x^{3} - 2)}^{\frac{1}{5}} + 241 - 15 {(x^{3} - 2)}^{\frac{4}{5}} - 180 {(x^{3} - 2)}^{\frac{3}{5}} - 810 {(x^{3} - 2)}^{\frac{2}{5}} - 1620 {(x^{3} - 2)}^{\frac{1}{5}} - 1215 + 90 {(x^{3} - 2)}^{\frac{3}{5}} + 810 {(x^{3} - 2)}^{\frac{2}{5}} + 2430 {(x^{3} - 2)}^{\frac{1}{5}} + 2430 - 270 {(x^{3} - 2)}^{\frac{2}{5}} - 1620 {(x^{3} - 2)}^{\frac{1}{5}} - 2430 + 405 {(x^{3} - 2)}^{\frac{1}{5}} + 1215 - 241}$

解题步骤 5.2.18

从 $15 {(x^{3} - 2)}^{\frac{4}{5}}$ 中减去 $15 {(x^{3} - 2)}^{\frac{4}{5}}$ 。

$f^{- 1} ({(x^{3} - 2)}^{\frac{1}{5}} + 3) = \sqrt[3]{x^{3} + 90 {(x^{3} - 2)}^{\frac{3}{5}} + 270 {(x^{3} - 2)}^{\frac{2}{5}} + 405 {(x^{3} - 2)}^{\frac{1}{5}} + 241 + 0 - 180 {(x^{3} - 2)}^{\frac{3}{5}} - 810 {(x^{3} - 2)}^{\frac{2}{5}} - 1620 {(x^{3} - 2)}^{\frac{1}{5}} - 1215 + 90 {(x^{3} - 2)}^{\frac{3}{5}} + 810 {(x^{3} - 2)}^{\frac{2}{5}} + 2430 {(x^{3} - 2)}^{\frac{1}{5}} + 2430 - 270 {(x^{3} - 2)}^{\frac{2}{5}} - 1620 {(x^{3} - 2)}^{\frac{1}{5}} - 2430 + 405 {(x^{3} - 2)}^{\frac{1}{5}} + 1215 - 241}$

解题步骤 5.2.19

将 $x^{3}$ 和 $0$ 相加。

$f^{- 1} ({(x^{3} - 2)}^{\frac{1}{5}} + 3) = \sqrt[3]{x^{3} + 90 {(x^{3} - 2)}^{\frac{3}{5}} + 270 {(x^{3} - 2)}^{\frac{2}{5}} + 405 {(x^{3} - 2)}^{\frac{1}{5}} + 241 - 180 {(x^{3} - 2)}^{\frac{3}{5}} - 810 {(x^{3} - 2)}^{\frac{2}{5}} - 1620 {(x^{3} - 2)}^{\frac{1}{5}} - 1215 + 90 {(x^{3} - 2)}^{\frac{3}{5}} + 810 {(x^{3} - 2)}^{\frac{2}{5}} + 2430 {(x^{3} - 2)}^{\frac{1}{5}} + 2430 - 270 {(x^{3} - 2)}^{\frac{2}{5}} - 1620 {(x^{3} - 2)}^{\frac{1}{5}} - 2430 + 405 {(x^{3} - 2)}^{\frac{1}{5}} + 1215 - 241}$

解题步骤 5.2.20

从 $90 {(x^{3} - 2)}^{\frac{3}{5}}$ 中减去 $180 {(x^{3} - 2)}^{\frac{3}{5}}$ 。

$f^{- 1} ({(x^{3} - 2)}^{\frac{1}{5}} + 3) = \sqrt[3]{x^{3} + 270 {(x^{3} - 2)}^{\frac{2}{5}} + 405 {(x^{3} - 2)}^{\frac{1}{5}} + 241 - 90 {(x^{3} - 2)}^{\frac{3}{5}} - 810 {(x^{3} - 2)}^{\frac{2}{5}} - 1620 {(x^{3} - 2)}^{\frac{1}{5}} - 1215 + 90 {(x^{3} - 2)}^{\frac{3}{5}} + 810 {(x^{3} - 2)}^{\frac{2}{5}} + 2430 {(x^{3} - 2)}^{\frac{1}{5}} + 2430 - 270 {(x^{3} - 2)}^{\frac{2}{5}} - 1620 {(x^{3} - 2)}^{\frac{1}{5}} - 2430 + 405 {(x^{3} - 2)}^{\frac{1}{5}} + 1215 - 241}$

解题步骤 5.2.21

从 $270 {(x^{3} - 2)}^{\frac{2}{5}}$ 中减去 $810 {(x^{3} - 2)}^{\frac{2}{5}}$ 。

$f^{- 1} ({(x^{3} - 2)}^{\frac{1}{5}} + 3) = \sqrt[3]{x^{3} + 405 {(x^{3} - 2)}^{\frac{1}{5}} + 241 - 90 {(x^{3} - 2)}^{\frac{3}{5}} - 540 {(x^{3} - 2)}^{\frac{2}{5}} - 1620 {(x^{3} - 2)}^{\frac{1}{5}} - 1215 + 90 {(x^{3} - 2)}^{\frac{3}{5}} + 810 {(x^{3} - 2)}^{\frac{2}{5}} + 2430 {(x^{3} - 2)}^{\frac{1}{5}} + 2430 - 270 {(x^{3} - 2)}^{\frac{2}{5}} - 1620 {(x^{3} - 2)}^{\frac{1}{5}} - 2430 + 405 {(x^{3} - 2)}^{\frac{1}{5}} + 1215 - 241}$

解题步骤 5.2.22

从 $405 {(x^{3} - 2)}^{\frac{1}{5}}$ 中减去 $1620 {(x^{3} - 2)}^{\frac{1}{5}}$ 。

$f^{- 1} ({(x^{3} - 2)}^{\frac{1}{5}} + 3) = \sqrt[3]{x^{3} + 241 - 90 {(x^{3} - 2)}^{\frac{3}{5}} - 540 {(x^{3} - 2)}^{\frac{2}{5}} - 1215 {(x^{3} - 2)}^{\frac{1}{5}} - 1215 + 90 {(x^{3} - 2)}^{\frac{3}{5}} + 810 {(x^{3} - 2)}^{\frac{2}{5}} + 2430 {(x^{3} - 2)}^{\frac{1}{5}} + 2430 - 270 {(x^{3} - 2)}^{\frac{2}{5}} - 1620 {(x^{3} - 2)}^{\frac{1}{5}} - 2430 + 405 {(x^{3} - 2)}^{\frac{1}{5}} + 1215 - 241}$

解题步骤 5.2.23

通过相加和相减进行化简。

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解题步骤 5.2.23.1

从 $241$ 中减去 $1215$ 。

$f^{- 1} ({(x^{3} - 2)}^{\frac{1}{5}} + 3) = \sqrt[3]{x^{3} - 974 - 90 {(x^{3} - 2)}^{\frac{3}{5}} - 540 {(x^{3} - 2)}^{\frac{2}{5}} - 1215 {(x^{3} - 2)}^{\frac{1}{5}} + 90 {(x^{3} - 2)}^{\frac{3}{5}} + 810 {(x^{3} - 2)}^{\frac{2}{5}} + 2430 {(x^{3} - 2)}^{\frac{1}{5}} + 2430 - 270 {(x^{3} - 2)}^{\frac{2}{5}} - 1620 {(x^{3} - 2)}^{\frac{1}{5}} - 2430 + 405 {(x^{3} - 2)}^{\frac{1}{5}} + 1215 - 241}$

解题步骤 5.2.23.2

将 $- 974$ 和 $2430$ 相加。

$f^{- 1} ({(x^{3} - 2)}^{\frac{1}{5}} + 3) = \sqrt[3]{x^{3} + 1456 - 90 {(x^{3} - 2)}^{\frac{3}{5}} - 540 {(x^{3} - 2)}^{\frac{2}{5}} - 1215 {(x^{3} - 2)}^{\frac{1}{5}} + 90 {(x^{3} - 2)}^{\frac{3}{5}} + 810 {(x^{3} - 2)}^{\frac{2}{5}} + 2430 {(x^{3} - 2)}^{\frac{1}{5}} - 270 {(x^{3} - 2)}^{\frac{2}{5}} - 1620 {(x^{3} - 2)}^{\frac{1}{5}} - 2430 + 405 {(x^{3} - 2)}^{\frac{1}{5}} + 1215 - 241}$

解题步骤 5.2.23.3

从 $1456$ 中减去 $2430$ 。

$f^{- 1} ({(x^{3} - 2)}^{\frac{1}{5}} + 3) = \sqrt[3]{x^{3} - 974 - 90 {(x^{3} - 2)}^{\frac{3}{5}} - 540 {(x^{3} - 2)}^{\frac{2}{5}} - 1215 {(x^{3} - 2)}^{\frac{1}{5}} + 90 {(x^{3} - 2)}^{\frac{3}{5}} + 810 {(x^{3} - 2)}^{\frac{2}{5}} + 2430 {(x^{3} - 2)}^{\frac{1}{5}} - 270 {(x^{3} - 2)}^{\frac{2}{5}} - 1620 {(x^{3} - 2)}^{\frac{1}{5}} + 405 {(x^{3} - 2)}^{\frac{1}{5}} + 1215 - 241}$

解题步骤 5.2.23.4

将 $- 974$ 和 $1215$ 相加。

$f^{- 1} ({(x^{3} - 2)}^{\frac{1}{5}} + 3) = \sqrt[3]{x^{3} + 241 - 90 {(x^{3} - 2)}^{\frac{3}{5}} - 540 {(x^{3} - 2)}^{\frac{2}{5}} - 1215 {(x^{3} - 2)}^{\frac{1}{5}} + 90 {(x^{3} - 2)}^{\frac{3}{5}} + 810 {(x^{3} - 2)}^{\frac{2}{5}} + 2430 {(x^{3} - 2)}^{\frac{1}{5}} - 270 {(x^{3} - 2)}^{\frac{2}{5}} - 1620 {(x^{3} - 2)}^{\frac{1}{5}} + 405 {(x^{3} - 2)}^{\frac{1}{5}} - 241}$

解题步骤 5.2.23.5

从 $241$ 中减去 $241$ 。

$f^{- 1} ({(x^{3} - 2)}^{\frac{1}{5}} + 3) = \sqrt[3]{x^{3} + 0 - 90 {(x^{3} - 2)}^{\frac{3}{5}} - 540 {(x^{3} - 2)}^{\frac{2}{5}} - 1215 {(x^{3} - 2)}^{\frac{1}{5}} + 90 {(x^{3} - 2)}^{\frac{3}{5}} + 810 {(x^{3} - 2)}^{\frac{2}{5}} + 2430 {(x^{3} - 2)}^{\frac{1}{5}} - 270 {(x^{3} - 2)}^{\frac{2}{5}} - 1620 {(x^{3} - 2)}^{\frac{1}{5}} + 405 {(x^{3} - 2)}^{\frac{1}{5}}}$

解题步骤 5.2.23.6

将 $x^{3}$ 和 $0$ 相加。

$f^{- 1} ({(x^{3} - 2)}^{\frac{1}{5}} + 3) = \sqrt[3]{x^{3} - 90 {(x^{3} - 2)}^{\frac{3}{5}} - 540 {(x^{3} - 2)}^{\frac{2}{5}} - 1215 {(x^{3} - 2)}^{\frac{1}{5}} + 90 {(x^{3} - 2)}^{\frac{3}{5}} + 810 {(x^{3} - 2)}^{\frac{2}{5}} + 2430 {(x^{3} - 2)}^{\frac{1}{5}} - 270 {(x^{3} - 2)}^{\frac{2}{5}} - 1620 {(x^{3} - 2)}^{\frac{1}{5}} + 405 {(x^{3} - 2)}^{\frac{1}{5}}}$

解题步骤 5.2.24

将 $- 90 {(x^{3} - 2)}^{\frac{3}{5}}$ 和 $90 {(x^{3} - 2)}^{\frac{3}{5}}$ 相加。

$f^{- 1} ({(x^{3} - 2)}^{\frac{1}{5}} + 3) = \sqrt[3]{x^{3} + 0 - 540 {(x^{3} - 2)}^{\frac{2}{5}} - 1215 {(x^{3} - 2)}^{\frac{1}{5}} + 810 {(x^{3} - 2)}^{\frac{2}{5}} + 2430 {(x^{3} - 2)}^{\frac{1}{5}} - 270 {(x^{3} - 2)}^{\frac{2}{5}} - 1620 {(x^{3} - 2)}^{\frac{1}{5}} + 405 {(x^{3} - 2)}^{\frac{1}{5}}}$

解题步骤 5.2.25

将 $x^{3}$ 和 $0$ 相加。

$f^{- 1} ({(x^{3} - 2)}^{\frac{1}{5}} + 3) = \sqrt[3]{x^{3} - 540 {(x^{3} - 2)}^{\frac{2}{5}} - 1215 {(x^{3} - 2)}^{\frac{1}{5}} + 810 {(x^{3} - 2)}^{\frac{2}{5}} + 2430 {(x^{3} - 2)}^{\frac{1}{5}} - 270 {(x^{3} - 2)}^{\frac{2}{5}} - 1620 {(x^{3} - 2)}^{\frac{1}{5}} + 405 {(x^{3} - 2)}^{\frac{1}{5}}}$

解题步骤 5.2.26

将 $- 540 {(x^{3} - 2)}^{\frac{2}{5}}$ 和 $810 {(x^{3} - 2)}^{\frac{2}{5}}$ 相加。

$f^{- 1} ({(x^{3} - 2)}^{\frac{1}{5}} + 3) = \sqrt[3]{x^{3} + 270 {(x^{3} - 2)}^{\frac{2}{5}} - 1215 {(x^{3} - 2)}^{\frac{1}{5}} + 2430 {(x^{3} - 2)}^{\frac{1}{5}} - 270 {(x^{3} - 2)}^{\frac{2}{5}} - 1620 {(x^{3} - 2)}^{\frac{1}{5}} + 405 {(x^{3} - 2)}^{\frac{1}{5}}}$

解题步骤 5.2.27

从 $270 {(x^{3} - 2)}^{\frac{2}{5}}$ 中减去 $270 {(x^{3} - 2)}^{\frac{2}{5}}$ 。

$f^{- 1} ({(x^{3} - 2)}^{\frac{1}{5}} + 3) = \sqrt[3]{x^{3} + 0 - 1215 {(x^{3} - 2)}^{\frac{1}{5}} + 2430 {(x^{3} - 2)}^{\frac{1}{5}} - 1620 {(x^{3} - 2)}^{\frac{1}{5}} + 405 {(x^{3} - 2)}^{\frac{1}{5}}}$

解题步骤 5.2.28

将 $x^{3}$ 和 $0$ 相加。

$f^{- 1} ({(x^{3} - 2)}^{\frac{1}{5}} + 3) = \sqrt[3]{x^{3} - 1215 {(x^{3} - 2)}^{\frac{1}{5}} + 2430 {(x^{3} - 2)}^{\frac{1}{5}} - 1620 {(x^{3} - 2)}^{\frac{1}{5}} + 405 {(x^{3} - 2)}^{\frac{1}{5}}}$

解题步骤 5.2.29

将 $- 1215 {(x^{3} - 2)}^{\frac{1}{5}}$ 和 $2430 {(x^{3} - 2)}^{\frac{1}{5}}$ 相加。

$f^{- 1} ({(x^{3} - 2)}^{\frac{1}{5}} + 3) = \sqrt[3]{x^{3} + 1215 {(x^{3} - 2)}^{\frac{1}{5}} - 1620 {(x^{3} - 2)}^{\frac{1}{5}} + 405 {(x^{3} - 2)}^{\frac{1}{5}}}$

解题步骤 5.2.30

从 $1215 {(x^{3} - 2)}^{\frac{1}{5}}$ 中减去 $1620 {(x^{3} - 2)}^{\frac{1}{5}}$ 。

$f^{- 1} ({(x^{3} - 2)}^{\frac{1}{5}} + 3) = \sqrt[3]{x^{3} - 405 {(x^{3} - 2)}^{\frac{1}{5}} + 405 {(x^{3} - 2)}^{\frac{1}{5}}}$

解题步骤 5.2.31

将 $- 405 {(x^{3} - 2)}^{\frac{1}{5}}$ 和 $405 {(x^{3} - 2)}^{\frac{1}{5}}$ 相加。

$f^{- 1} ({(x^{3} - 2)}^{\frac{1}{5}} + 3) = \sqrt[3]{x^{3} + 0}$

解题步骤 5.2.32

将 $x^{3}$ 和 $0$ 相加。

$f^{- 1} ({(x^{3} - 2)}^{\frac{1}{5}} + 3) = \sqrt[3]{x^{3}}$

解题步骤 5.2.33

假设各项均为实数，将其从根式下提取出来。

$f^{- 1} ({(x^{3} - 2)}^{\frac{1}{5}} + 3) = x$

解题步骤 5.3

计算 $f (f^{- 1} (x))$ 。

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解题步骤 5.3.1

建立复合结果函数。

$f (f^{- 1} (x))$

解题步骤 5.3.2

通过将 $f^{- 1}$ 的值代入 $f$ 来计算 $f (\sqrt[3]{x^{5} - 15 x^{4} + 90 x^{3} - 270 x^{2} + 405 x - 241})$ 。

$f (\sqrt[3]{x^{5} - 15 x^{4} + 90 x^{3} - 270 x^{2} + 405 x - 241}) = {({(\sqrt[3]{x^{5} - 15 x^{4} + 90 x^{3} - 270 x^{2} + 405 x - 241})}^{3} - 2)}^{\frac{1}{5}} + 3$

解题步骤 5.3.3

化简每一项。

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解题步骤 5.3.3.1

将 ${\sqrt[3]{x^{5} - 15 x^{4} + 90 x^{3} - 270 x^{2} + 405 x - 241}}^{3}$ 重写为 $x^{5} - 15 x^{4} + 90 x^{3} - 270 x^{2} + 405 x - 241$ 。

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解题步骤 5.3.3.1.1

使用 $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ ，将 $\sqrt[3]{x^{5} - 15 x^{4} + 90 x^{3} - 270 x^{2} + 405 x - 241}$ 重写成 ${(x^{5} - 15 x^{4} + 90 x^{3} - 270 x^{2} + 405 x - 241)}^{\frac{1}{3}}$ 。

$f (\sqrt[3]{x^{5} - 15 x^{4} + 90 x^{3} - 270 x^{2} + 405 x - 241}) = {({({(x^{5} - 15 x^{4} + 90 x^{3} - 270 x^{2} + 405 x - 241)}^{\frac{1}{3}})}^{3} - 2)}^{\frac{1}{5}} + 3$

解题步骤 5.3.3.1.2

运用幂法则并将指数相乘， ${(a^{m})}^{n} = a^{m n}$ 。

$f (\sqrt[3]{x^{5} - 15 x^{4} + 90 x^{3} - 270 x^{2} + 405 x - 241}) = {({(x^{5} - 15 x^{4} + 90 x^{3} - 270 x^{2} + 405 x - 241)}^{\frac{1}{3} \cdot 3} - 2)}^{\frac{1}{5}} + 3$

解题步骤 5.3.3.1.3

组合 $\frac{1}{3}$ 和 $3$ 。

$f (\sqrt[3]{x^{5} - 15 x^{4} + 90 x^{3} - 270 x^{2} + 405 x - 241}) = {({(x^{5} - 15 x^{4} + 90 x^{3} - 270 x^{2} + 405 x - 241)}^{\frac{3}{3}} - 2)}^{\frac{1}{5}} + 3$

解题步骤 5.3.3.1.4

约去 $3$ 的公因数。

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解题步骤 5.3.3.1.4.1

约去公因数。

$f (\sqrt[3]{x^{5} - 15 x^{4} + 90 x^{3} - 270 x^{2} + 405 x - 241}) = {({(x^{5} - 15 x^{4} + 90 x^{3} - 270 x^{2} + 405 x - 241)}^{\frac{3}{3}} - 2)}^{\frac{1}{5}} + 3$

解题步骤 5.3.3.1.4.2

重写表达式。

$f (\sqrt[3]{x^{5} - 15 x^{4} + 90 x^{3} - 270 x^{2} + 405 x - 241}) = {((x^{5} - 15 x^{4} + 90 x^{3} - 270 x^{2} + 405 x - 241) - 2)}^{\frac{1}{5}} + 3$

解题步骤 5.3.3.1.5

化简。

$f (\sqrt[3]{x^{5} - 15 x^{4} + 90 x^{3} - 270 x^{2} + 405 x - 241}) = {(x^{5} - 15 x^{4} + 90 x^{3} - 270 x^{2} + 405 x - 241 - 2)}^{\frac{1}{5}} + 3$

解题步骤 5.3.3.2

从 $- 241$ 中减去 $2$ 。

$f (\sqrt[3]{x^{5} - 15 x^{4} + 90 x^{3} - 270 x^{2} + 405 x - 241}) = {(x^{5} - 15 x^{4} + 90 x^{3} - 270 x^{2} + 405 x - 243)}^{\frac{1}{5}} + 3$

解题步骤 5.4

由于 $f^{- 1} (f (x)) = x$ 和 $f (f^{- 1} (x)) = x$ ，因此 $f^{- 1} (x) = \sqrt[3]{x^{5} - 15 x^{4} + 90 x^{3} - 270 x^{2} + 405 x - 241}$ 为 $f (x) = {(x^{3} - 2)}^{\frac{1}{5}} + 3$ 的反函数。

$f^{- 1} (x) = \sqrt[3]{x^{5} - 15 x^{4} + 90 x^{3} - 270 x^{2} + 405 x - 241}$

代数 示例

代数示例