初级微积分示例

$f (x) = 2 x^{\frac{1}{5}} - 2$ ; find $f^{- 1} (x)$

解题步骤 1

求 $f^{- 1} (x)$ 。

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解题步骤 1.1

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

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

解题步骤 1.2

交换变量。

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

解题步骤 1.3

求解 $y$ 。

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解题步骤 1.3.1

将方程重写为 $2 y^{\frac{1}{5}} - 2 = x$ 。

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

解题步骤 1.3.2

将所有不包含 $y$ 的项移到等式右边。

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解题步骤 1.3.2.1

在等式两边都加上 $2$ 。

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

解题步骤 1.3.2.2

在等式两边都加上 $x$ 。

$2 y^{\frac{1}{5}} = 2 + x$

解题步骤 1.3.3

将方程两边同时进行 $5$ 次方运算以消去左边的分数指数。

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

解题步骤 1.3.4

化简指数。

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解题步骤 1.3.4.1

化简左边。

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解题步骤 1.3.4.1.1

化简 ${(2 y^{\frac{1}{5}})}^{5}$ 。

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解题步骤 1.3.4.1.1.1

对 $2 y^{\frac{1}{5}}$ 运用乘积法则。

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

解题步骤 1.3.4.1.1.2

对 $2$ 进行 $5$ 次方运算。

$32 {(y^{\frac{1}{5}})}^{5} = {(2 + x)}^{5}$

解题步骤 1.3.4.1.1.3

将 ${(y^{\frac{1}{5}})}^{5}$ 中的指数相乘。

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解题步骤 1.3.4.1.1.3.1

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

$32 y^{\frac{1}{5} \cdot 5} = {(2 + x)}^{5}$

解题步骤 1.3.4.1.1.3.2

约去 $5$ 的公因数。

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解题步骤 1.3.4.1.1.3.2.1

约去公因数。

$32 y^{\frac{1}{5} \cdot 5} = {(2 + x)}^{5}$

解题步骤 1.3.4.1.1.3.2.2

重写表达式。

$32 y^{1} = {(2 + x)}^{5}$

解题步骤 1.3.4.1.1.4

化简。

$32 y = {(2 + x)}^{5}$

解题步骤 1.3.4.2

化简右边。

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解题步骤 1.3.4.2.1

化简 ${(2 + x)}^{5}$ 。

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解题步骤 1.3.4.2.1.1

使用二项式定理。

$32 y = 2^{5} + 5 \cdot 2^{4} x + 10 \cdot 2^{3} x^{2} + 10 \cdot 2^{2} x^{3} + 5 \cdot 2 x^{4} + x^{5}$

解题步骤 1.3.4.2.1.2

化简每一项。

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解题步骤 1.3.4.2.1.2.1

对 $2$ 进行 $5$ 次方运算。

$32 y = 32 + 5 \cdot 2^{4} x + 10 \cdot 2^{3} x^{2} + 10 \cdot 2^{2} x^{3} + 5 \cdot 2 x^{4} + x^{5}$

解题步骤 1.3.4.2.1.2.2

对 $2$ 进行 $4$ 次方运算。

$32 y = 32 + 5 \cdot 16 x + 10 \cdot 2^{3} x^{2} + 10 \cdot 2^{2} x^{3} + 5 \cdot 2 x^{4} + x^{5}$

解题步骤 1.3.4.2.1.2.3

将 $5$ 乘以 $16$ 。

$32 y = 32 + 80 x + 10 \cdot 2^{3} x^{2} + 10 \cdot 2^{2} x^{3} + 5 \cdot 2 x^{4} + x^{5}$

解题步骤 1.3.4.2.1.2.4

对 $2$ 进行 $3$ 次方运算。

$32 y = 32 + 80 x + 10 \cdot 8 x^{2} + 10 \cdot 2^{2} x^{3} + 5 \cdot 2 x^{4} + x^{5}$

解题步骤 1.3.4.2.1.2.5

将 $10$ 乘以 $8$ 。

$32 y = 32 + 80 x + 80 x^{2} + 10 \cdot 2^{2} x^{3} + 5 \cdot 2 x^{4} + x^{5}$

解题步骤 1.3.4.2.1.2.6

对 $2$ 进行 $2$ 次方运算。

$32 y = 32 + 80 x + 80 x^{2} + 10 \cdot 4 x^{3} + 5 \cdot 2 x^{4} + x^{5}$

解题步骤 1.3.4.2.1.2.7

将 $10$ 乘以 $4$ 。

$32 y = 32 + 80 x + 80 x^{2} + 40 x^{3} + 5 \cdot 2 x^{4} + x^{5}$

解题步骤 1.3.4.2.1.2.8

将 $5$ 乘以 $2$ 。

$32 y = 32 + 80 x + 80 x^{2} + 40 x^{3} + 10 x^{4} + x^{5}$

解题步骤 1.3.5

将 $32 y = 32 + 80 x + 80 x^{2} + 40 x^{3} + 10 x^{4} + x^{5}$ 中的每一项除以 $32$ 并化简。

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解题步骤 1.3.5.1

将 $32 y = 32 + 80 x + 80 x^{2} + 40 x^{3} + 10 x^{4} + x^{5}$ 中的每一项都除以 $32$ 。

$\frac{32 y}{32} = \frac{32}{32} + \frac{80 x}{32} + \frac{80 x^{2}}{32} + \frac{40 x^{3}}{32} + \frac{10 x^{4}}{32} + \frac{x^{5}}{32}$

解题步骤 1.3.5.2

化简左边。

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解题步骤 1.3.5.2.1

约去 $32$ 的公因数。

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解题步骤 1.3.5.2.1.1

约去公因数。

$\frac{32 y}{32} = \frac{32}{32} + \frac{80 x}{32} + \frac{80 x^{2}}{32} + \frac{40 x^{3}}{32} + \frac{10 x^{4}}{32} + \frac{x^{5}}{32}$

解题步骤 1.3.5.2.1.2

用 $y$ 除以 $1$ 。

$y = \frac{32}{32} + \frac{80 x}{32} + \frac{80 x^{2}}{32} + \frac{40 x^{3}}{32} + \frac{10 x^{4}}{32} + \frac{x^{5}}{32}$

解题步骤 1.3.5.3

化简右边。

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解题步骤 1.3.5.3.1

化简每一项。

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解题步骤 1.3.5.3.1.1

用 $32$ 除以 $32$ 。

$y = 1 + \frac{80 x}{32} + \frac{80 x^{2}}{32} + \frac{40 x^{3}}{32} + \frac{10 x^{4}}{32} + \frac{x^{5}}{32}$

解题步骤 1.3.5.3.1.2

约去 $80$ 和 $32$ 的公因数。

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解题步骤 1.3.5.3.1.2.1

从 $80 x$ 中分解出因数 $16$ 。

$y = 1 + \frac{16 (5 x)}{32} + \frac{80 x^{2}}{32} + \frac{40 x^{3}}{32} + \frac{10 x^{4}}{32} + \frac{x^{5}}{32}$

解题步骤 1.3.5.3.1.2.2

约去公因数。

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解题步骤 1.3.5.3.1.2.2.1

从 $32$ 中分解出因数 $16$ 。

$y = 1 + \frac{16 (5 x)}{16 (2)} + \frac{80 x^{2}}{32} + \frac{40 x^{3}}{32} + \frac{10 x^{4}}{32} + \frac{x^{5}}{32}$

解题步骤 1.3.5.3.1.2.2.2

约去公因数。

$y = 1 + \frac{16 (5 x)}{16 \cdot 2} + \frac{80 x^{2}}{32} + \frac{40 x^{3}}{32} + \frac{10 x^{4}}{32} + \frac{x^{5}}{32}$

解题步骤 1.3.5.3.1.2.2.3

重写表达式。

$y = 1 + \frac{5 x}{2} + \frac{80 x^{2}}{32} + \frac{40 x^{3}}{32} + \frac{10 x^{4}}{32} + \frac{x^{5}}{32}$

解题步骤 1.3.5.3.1.3

约去 $80$ 和 $32$ 的公因数。

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解题步骤 1.3.5.3.1.3.1

从 $80 x^{2}$ 中分解出因数 $16$ 。

$y = 1 + \frac{5 x}{2} + \frac{16 (5 x^{2})}{32} + \frac{40 x^{3}}{32} + \frac{10 x^{4}}{32} + \frac{x^{5}}{32}$

解题步骤 1.3.5.3.1.3.2

约去公因数。

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解题步骤 1.3.5.3.1.3.2.1

从 $32$ 中分解出因数 $16$ 。

$y = 1 + \frac{5 x}{2} + \frac{16 (5 x^{2})}{16 (2)} + \frac{40 x^{3}}{32} + \frac{10 x^{4}}{32} + \frac{x^{5}}{32}$

解题步骤 1.3.5.3.1.3.2.2

约去公因数。

$y = 1 + \frac{5 x}{2} + \frac{16 (5 x^{2})}{16 \cdot 2} + \frac{40 x^{3}}{32} + \frac{10 x^{4}}{32} + \frac{x^{5}}{32}$

解题步骤 1.3.5.3.1.3.2.3

重写表达式。

$y = 1 + \frac{5 x}{2} + \frac{5 x^{2}}{2} + \frac{40 x^{3}}{32} + \frac{10 x^{4}}{32} + \frac{x^{5}}{32}$

解题步骤 1.3.5.3.1.4

约去 $40$ 和 $32$ 的公因数。

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解题步骤 1.3.5.3.1.4.1

从 $40 x^{3}$ 中分解出因数 $8$ 。

$y = 1 + \frac{5 x}{2} + \frac{5 x^{2}}{2} + \frac{8 (5 x^{3})}{32} + \frac{10 x^{4}}{32} + \frac{x^{5}}{32}$

解题步骤 1.3.5.3.1.4.2

约去公因数。

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解题步骤 1.3.5.3.1.4.2.1

从 $32$ 中分解出因数 $8$ 。

$y = 1 + \frac{5 x}{2} + \frac{5 x^{2}}{2} + \frac{8 (5 x^{3})}{8 (4)} + \frac{10 x^{4}}{32} + \frac{x^{5}}{32}$

解题步骤 1.3.5.3.1.4.2.2

约去公因数。

$y = 1 + \frac{5 x}{2} + \frac{5 x^{2}}{2} + \frac{8 (5 x^{3})}{8 \cdot 4} + \frac{10 x^{4}}{32} + \frac{x^{5}}{32}$

解题步骤 1.3.5.3.1.4.2.3

重写表达式。

$y = 1 + \frac{5 x}{2} + \frac{5 x^{2}}{2} + \frac{5 x^{3}}{4} + \frac{10 x^{4}}{32} + \frac{x^{5}}{32}$

解题步骤 1.3.5.3.1.5

约去 $10$ 和 $32$ 的公因数。

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解题步骤 1.3.5.3.1.5.1

从 $10 x^{4}$ 中分解出因数 $2$ 。

$y = 1 + \frac{5 x}{2} + \frac{5 x^{2}}{2} + \frac{5 x^{3}}{4} + \frac{2 (5 x^{4})}{32} + \frac{x^{5}}{32}$

解题步骤 1.3.5.3.1.5.2

约去公因数。

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解题步骤 1.3.5.3.1.5.2.1

从 $32$ 中分解出因数 $2$ 。

$y = 1 + \frac{5 x}{2} + \frac{5 x^{2}}{2} + \frac{5 x^{3}}{4} + \frac{2 (5 x^{4})}{2 (16)} + \frac{x^{5}}{32}$

解题步骤 1.3.5.3.1.5.2.2

约去公因数。

$y = 1 + \frac{5 x}{2} + \frac{5 x^{2}}{2} + \frac{5 x^{3}}{4} + \frac{2 (5 x^{4})}{2 \cdot 16} + \frac{x^{5}}{32}$

解题步骤 1.3.5.3.1.5.2.3

重写表达式。

$y = 1 + \frac{5 x}{2} + \frac{5 x^{2}}{2} + \frac{5 x^{3}}{4} + \frac{5 x^{4}}{16} + \frac{x^{5}}{32}$

解题步骤 1.4

使用 $f^{- 1} (x)$ 替换 $y$ ，以得到最终答案。

$f^{- 1} (x) = 1 + \frac{5 x}{2} + \frac{5 x^{2}}{2} + \frac{5 x^{3}}{4} + \frac{5 x^{4}}{16} + \frac{x^{5}}{32}$

解题步骤 1.5

验证 $f^{- 1} (x) = 1 + \frac{5 x}{2} + \frac{5 x^{2}}{2} + \frac{5 x^{3}}{4} + \frac{5 x^{4}}{16} + \frac{x^{5}}{32}$ 是否为 $f (x) = 2 x^{\frac{1}{5}} - 2$ 的反函数。

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解题步骤 1.5.1

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 1.5.2

计算 $f^{- 1} (f (x))$ 。

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解题步骤 1.5.2.1

建立复合结果函数。

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

解题步骤 1.5.2.2

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

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + \frac{5 (2 x^{\frac{1}{5}} - 2)}{2} + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{2}}{2} + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{3}}{4} + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{4}}{16} + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

解题步骤 1.5.2.3

化简每一项。

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解题步骤 1.5.2.3.1

从 $5 (2 x^{\frac{1}{5}} - 2)$ 中分解出因数 $2$ 。

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + \frac{2 (5 (x^{\frac{1}{5}} - 1))}{2} + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{2}}{2} + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{3}}{4} + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{4}}{16} + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

解题步骤 1.5.2.3.2

约去公因数。

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解题步骤 1.5.2.3.2.1

从 $2$ 中分解出因数 $2$ 。

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + \frac{2 (5 (x^{\frac{1}{5}} - 1))}{2 (1)} + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{2}}{2} + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{3}}{4} + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{4}}{16} + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

解题步骤 1.5.2.3.2.2

约去公因数。

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + \frac{2 (5 (x^{\frac{1}{5}} - 1))}{2 \cdot 1} + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{2}}{2} + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{3}}{4} + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{4}}{16} + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

解题步骤 1.5.2.3.2.3

重写表达式。

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + \frac{5 (x^{\frac{1}{5}} - 1)}{1} + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{2}}{2} + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{3}}{4} + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{4}}{16} + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

解题步骤 1.5.2.3.2.4

用 $5 (x^{\frac{1}{5}} - 1)$ 除以 $1$ 。

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 (x^{\frac{1}{5}} - 1) + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{2}}{2} + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{3}}{4} + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{4}}{16} + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

解题步骤 1.5.2.3.3

运用分配律。

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} + 5 \cdot - 1 + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{2}}{2} + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{3}}{4} + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{4}}{16} + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

解题步骤 1.5.2.3.4

将 $5$ 乘以 $- 1$ 。

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{2}}{2} + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{3}}{4} + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{4}}{16} + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

解题步骤 1.5.2.3.5

化简分子。

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解题步骤 1.5.2.3.5.1

从 $2 x^{\frac{1}{5}} - 2$ 中分解出因数 $2$ 。

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解题步骤 1.5.2.3.5.1.1

从 $2 x^{\frac{1}{5}}$ 中分解出因数 $2$ 。

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + \frac{5 {(2 (x^{\frac{1}{5}}) - 2)}^{2}}{2} + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{3}}{4} + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{4}}{16} + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

解题步骤 1.5.2.3.5.1.2

从 $- 2$ 中分解出因数 $2$ 。

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + \frac{5 {(2 (x^{\frac{1}{5}}) + 2 (- 1))}^{2}}{2} + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{3}}{4} + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{4}}{16} + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

解题步骤 1.5.2.3.5.1.3

从 $2 (x^{\frac{1}{5}}) + 2 (- 1)$ 中分解出因数 $2$ 。

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + \frac{5 {(2 (x^{\frac{1}{5}} - 1))}^{2}}{2} + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{3}}{4} + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{4}}{16} + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

解题步骤 1.5.2.3.5.2

对 $2 (x^{\frac{1}{5}} - 1)$ 运用乘积法则。

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + \frac{5 \cdot (2^{2} {(x^{\frac{1}{5}} - 1)}^{2})}{2} + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{3}}{4} + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{4}}{16} + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

解题步骤 1.5.2.3.5.3

对 $2$ 进行 $2$ 次方运算。

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + \frac{5 \cdot (4 {(x^{\frac{1}{5}} - 1)}^{2})}{2} + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{3}}{4} + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{4}}{16} + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

解题步骤 1.5.2.3.5.4

将 $5$ 乘以 $4$ 。

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + \frac{20 {(x^{\frac{1}{5}} - 1)}^{2}}{2} + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{3}}{4} + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{4}}{16} + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

解题步骤 1.5.2.3.6

从 $20 {(x^{\frac{1}{5}} - 1)}^{2}$ 中分解出因数 $2$ 。

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + \frac{2 (10 {(x^{\frac{1}{5}} - 1)}^{2})}{2} + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{3}}{4} + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{4}}{16} + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

解题步骤 1.5.2.3.7

约去公因数。

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解题步骤 1.5.2.3.7.1

从 $2$ 中分解出因数 $2$ 。

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + \frac{2 (10 {(x^{\frac{1}{5}} - 1)}^{2})}{2 (1)} + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{3}}{4} + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{4}}{16} + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

解题步骤 1.5.2.3.7.2

约去公因数。

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + \frac{2 (10 {(x^{\frac{1}{5}} - 1)}^{2})}{2 \cdot 1} + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{3}}{4} + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{4}}{16} + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

解题步骤 1.5.2.3.7.3

重写表达式。

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + \frac{10 {(x^{\frac{1}{5}} - 1)}^{2}}{1} + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{3}}{4} + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{4}}{16} + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

解题步骤 1.5.2.3.7.4

用 $10 {(x^{\frac{1}{5}} - 1)}^{2}$ 除以 $1$ 。

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 {(x^{\frac{1}{5}} - 1)}^{2} + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{3}}{4} + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{4}}{16} + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

解题步骤 1.5.2.3.8

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

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 ((x^{\frac{1}{5}} - 1) (x^{\frac{1}{5}} - 1)) + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{3}}{4} + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{4}}{16} + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

解题步骤 1.5.2.3.9

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

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解题步骤 1.5.2.3.9.1

运用分配律。

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 (x^{\frac{1}{5}} (x^{\frac{1}{5}} - 1) - 1 (x^{\frac{1}{5}} - 1)) + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{3}}{4} + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{4}}{16} + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

解题步骤 1.5.2.3.9.2

运用分配律。

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 (x^{\frac{1}{5}} x^{\frac{1}{5}} + x^{\frac{1}{5}} \cdot - 1 - 1 (x^{\frac{1}{5}} - 1)) + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{3}}{4} + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{4}}{16} + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

解题步骤 1.5.2.3.9.3

运用分配律。

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 (x^{\frac{1}{5}} x^{\frac{1}{5}} + x^{\frac{1}{5}} \cdot - 1 - 1 x^{\frac{1}{5}} - 1 \cdot - 1) + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{3}}{4} + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{4}}{16} + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

解题步骤 1.5.2.3.10

化简并合并同类项。

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解题步骤 1.5.2.3.10.1

化简每一项。

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解题步骤 1.5.2.3.10.1.1

通过指数相加将 $x^{\frac{1}{5}}$ 乘以 $x^{\frac{1}{5}}$ 。

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解题步骤 1.5.2.3.10.1.1.1

使用幂法则 $a^{m} a^{n} = a^{m + n}$ 合并指数。

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 (x^{\frac{1}{5} + \frac{1}{5}} + x^{\frac{1}{5}} \cdot - 1 - 1 x^{\frac{1}{5}} - 1 \cdot - 1) + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{3}}{4} + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{4}}{16} + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

解题步骤 1.5.2.3.10.1.1.2

在公分母上合并分子。

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 (x^{\frac{1 + 1}{5}} + x^{\frac{1}{5}} \cdot - 1 - 1 x^{\frac{1}{5}} - 1 \cdot - 1) + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{3}}{4} + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{4}}{16} + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

解题步骤 1.5.2.3.10.1.1.3

将 $1$ 和 $1$ 相加。

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 (x^{\frac{2}{5}} + x^{\frac{1}{5}} \cdot - 1 - 1 x^{\frac{1}{5}} - 1 \cdot - 1) + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{3}}{4} + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{4}}{16} + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

解题步骤 1.5.2.3.10.1.2

将 $- 1$ 移到 $x^{\frac{1}{5}}$ 的左侧。

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 (x^{\frac{2}{5}} - 1 \cdot x^{\frac{1}{5}} - 1 x^{\frac{1}{5}} - 1 \cdot - 1) + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{3}}{4} + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{4}}{16} + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

解题步骤 1.5.2.3.10.1.3

将 $- 1 x^{\frac{1}{5}}$ 重写为 $- x^{\frac{1}{5}}$ 。

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 (x^{\frac{2}{5}} - x^{\frac{1}{5}} - 1 x^{\frac{1}{5}} - 1 \cdot - 1) + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{3}}{4} + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{4}}{16} + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

解题步骤 1.5.2.3.10.1.4

将 $- 1 x^{\frac{1}{5}}$ 重写为 $- x^{\frac{1}{5}}$ 。

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 (x^{\frac{2}{5}} - x^{\frac{1}{5}} - x^{\frac{1}{5}} - 1 \cdot - 1) + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{3}}{4} + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{4}}{16} + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

解题步骤 1.5.2.3.10.1.5

将 $- 1$ 乘以 $- 1$ 。

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 (x^{\frac{2}{5}} - x^{\frac{1}{5}} - x^{\frac{1}{5}} + 1) + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{3}}{4} + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{4}}{16} + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

解题步骤 1.5.2.3.10.2

从 $- x^{\frac{1}{5}}$ 中减去 $x^{\frac{1}{5}}$ 。

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 (x^{\frac{2}{5}} - 2 x^{\frac{1}{5}} + 1) + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{3}}{4} + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{4}}{16} + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

解题步骤 1.5.2.3.11

运用分配律。

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 x^{\frac{2}{5}} + 10 (- 2 x^{\frac{1}{5}}) + 10 \cdot 1 + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{3}}{4} + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{4}}{16} + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

解题步骤 1.5.2.3.12

化简。

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解题步骤 1.5.2.3.12.1

将 $- 2$ 乘以 $10$ 。

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 \cdot 1 + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{3}}{4} + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{4}}{16} + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

解题步骤 1.5.2.3.12.2

将 $10$ 乘以 $1$ 。

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{3}}{4} + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{4}}{16} + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

解题步骤 1.5.2.3.13

化简分子。

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解题步骤 1.5.2.3.13.1

从 $2 x^{\frac{1}{5}} - 2$ 中分解出因数 $2$ 。

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解题步骤 1.5.2.3.13.1.1

从 $2 x^{\frac{1}{5}}$ 中分解出因数 $2$ 。

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 + \frac{5 {(2 (x^{\frac{1}{5}}) - 2)}^{3}}{4} + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{4}}{16} + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

解题步骤 1.5.2.3.13.1.2

从 $- 2$ 中分解出因数 $2$ 。

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 + \frac{5 {(2 (x^{\frac{1}{5}}) + 2 (- 1))}^{3}}{4} + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{4}}{16} + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

解题步骤 1.5.2.3.13.1.3

从 $2 (x^{\frac{1}{5}}) + 2 (- 1)$ 中分解出因数 $2$ 。

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 + \frac{5 {(2 (x^{\frac{1}{5}} - 1))}^{3}}{4} + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{4}}{16} + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

解题步骤 1.5.2.3.13.2

对 $2 (x^{\frac{1}{5}} - 1)$ 运用乘积法则。

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 + \frac{5 \cdot (2^{3} {(x^{\frac{1}{5}} - 1)}^{3})}{4} + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{4}}{16} + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

解题步骤 1.5.2.3.13.3

对 $2$ 进行 $3$ 次方运算。

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 + \frac{5 \cdot (8 {(x^{\frac{1}{5}} - 1)}^{3})}{4} + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{4}}{16} + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

解题步骤 1.5.2.3.13.4

将 $5$ 乘以 $8$ 。

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 + \frac{40 {(x^{\frac{1}{5}} - 1)}^{3}}{4} + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{4}}{16} + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

解题步骤 1.5.2.3.14

从 $40 {(x^{\frac{1}{5}} - 1)}^{3}$ 中分解出因数 $4$ 。

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 + \frac{4 (10 {(x^{\frac{1}{5}} - 1)}^{3})}{4} + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{4}}{16} + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

解题步骤 1.5.2.3.15

约去公因数。

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解题步骤 1.5.2.3.15.1

从 $4$ 中分解出因数 $4$ 。

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 + \frac{4 (10 {(x^{\frac{1}{5}} - 1)}^{3})}{4 (1)} + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{4}}{16} + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

解题步骤 1.5.2.3.15.2

约去公因数。

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 + \frac{4 (10 {(x^{\frac{1}{5}} - 1)}^{3})}{4 \cdot 1} + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{4}}{16} + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

解题步骤 1.5.2.3.15.3

重写表达式。

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 + \frac{10 {(x^{\frac{1}{5}} - 1)}^{3}}{1} + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{4}}{16} + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

解题步骤 1.5.2.3.15.4

用 $10 {(x^{\frac{1}{5}} - 1)}^{3}$ 除以 $1$ 。

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 + 10 {(x^{\frac{1}{5}} - 1)}^{3} + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{4}}{16} + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

解题步骤 1.5.2.3.16

使用二项式定理。

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 + 10 ({(x^{\frac{1}{5}})}^{3} + 3 {(x^{\frac{1}{5}})}^{2} \cdot - 1 + 3 x^{\frac{1}{5}} {(- 1)}^{2} + {(- 1)}^{3}) + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{4}}{16} + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

解题步骤 1.5.2.3.17

化简每一项。

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解题步骤 1.5.2.3.17.1

将 ${(x^{\frac{1}{5}})}^{3}$ 中的指数相乘。

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解题步骤 1.5.2.3.17.1.1

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

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 + 10 (x^{\frac{1}{5} \cdot 3} + 3 {(x^{\frac{1}{5}})}^{2} \cdot - 1 + 3 x^{\frac{1}{5}} {(- 1)}^{2} + {(- 1)}^{3}) + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{4}}{16} + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

解题步骤 1.5.2.3.17.1.2

组合 $\frac{1}{5}$ 和 $3$ 。

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 + 10 (x^{\frac{3}{5}} + 3 {(x^{\frac{1}{5}})}^{2} \cdot - 1 + 3 x^{\frac{1}{5}} {(- 1)}^{2} + {(- 1)}^{3}) + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{4}}{16} + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

解题步骤 1.5.2.3.17.2

将 ${(x^{\frac{1}{5}})}^{2}$ 中的指数相乘。

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解题步骤 1.5.2.3.17.2.1

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

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 + 10 (x^{\frac{3}{5}} + 3 x^{\frac{1}{5} \cdot 2} \cdot - 1 + 3 x^{\frac{1}{5}} {(- 1)}^{2} + {(- 1)}^{3}) + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{4}}{16} + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

解题步骤 1.5.2.3.17.2.2

组合 $\frac{1}{5}$ 和 $2$ 。

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 + 10 (x^{\frac{3}{5}} + 3 x^{\frac{2}{5}} \cdot - 1 + 3 x^{\frac{1}{5}} {(- 1)}^{2} + {(- 1)}^{3}) + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{4}}{16} + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

解题步骤 1.5.2.3.17.3

将 $- 1$ 乘以 $3$ 。

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 + 10 (x^{\frac{3}{5}} - 3 x^{\frac{2}{5}} + 3 x^{\frac{1}{5}} {(- 1)}^{2} + {(- 1)}^{3}) + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{4}}{16} + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

解题步骤 1.5.2.3.17.4

对 $- 1$ 进行 $2$ 次方运算。

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 + 10 (x^{\frac{3}{5}} - 3 x^{\frac{2}{5}} + 3 x^{\frac{1}{5}} \cdot 1 + {(- 1)}^{3}) + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{4}}{16} + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

解题步骤 1.5.2.3.17.5

将 $3$ 乘以 $1$ 。

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 + 10 (x^{\frac{3}{5}} - 3 x^{\frac{2}{5}} + 3 x^{\frac{1}{5}} + {(- 1)}^{3}) + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{4}}{16} + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

解题步骤 1.5.2.3.17.6

对 $- 1$ 进行 $3$ 次方运算。

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 + 10 (x^{\frac{3}{5}} - 3 x^{\frac{2}{5}} + 3 x^{\frac{1}{5}} - 1) + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{4}}{16} + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

解题步骤 1.5.2.3.18

运用分配律。

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 + 10 x^{\frac{3}{5}} + 10 (- 3 x^{\frac{2}{5}}) + 10 (3 x^{\frac{1}{5}}) + 10 \cdot - 1 + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{4}}{16} + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

解题步骤 1.5.2.3.19

化简。

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解题步骤 1.5.2.3.19.1

将 $- 3$ 乘以 $10$ 。

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 + 10 x^{\frac{3}{5}} - 30 x^{\frac{2}{5}} + 10 (3 x^{\frac{1}{5}}) + 10 \cdot - 1 + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{4}}{16} + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

解题步骤 1.5.2.3.19.2

将 $3$ 乘以 $10$ 。

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 + 10 x^{\frac{3}{5}} - 30 x^{\frac{2}{5}} + 30 x^{\frac{1}{5}} + 10 \cdot - 1 + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{4}}{16} + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

解题步骤 1.5.2.3.19.3

将 $10$ 乘以 $- 1$ 。

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 + 10 x^{\frac{3}{5}} - 30 x^{\frac{2}{5}} + 30 x^{\frac{1}{5}} - 10 + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{4}}{16} + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

解题步骤 1.5.2.3.20

化简分子。

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解题步骤 1.5.2.3.20.1

从 $2 x^{\frac{1}{5}} - 2$ 中分解出因数 $2$ 。

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解题步骤 1.5.2.3.20.1.1

从 $2 x^{\frac{1}{5}}$ 中分解出因数 $2$ 。

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 + 10 x^{\frac{3}{5}} - 30 x^{\frac{2}{5}} + 30 x^{\frac{1}{5}} - 10 + \frac{5 {(2 (x^{\frac{1}{5}}) - 2)}^{4}}{16} + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

解题步骤 1.5.2.3.20.1.2

从 $- 2$ 中分解出因数 $2$ 。

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 + 10 x^{\frac{3}{5}} - 30 x^{\frac{2}{5}} + 30 x^{\frac{1}{5}} - 10 + \frac{5 {(2 (x^{\frac{1}{5}}) + 2 (- 1))}^{4}}{16} + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

解题步骤 1.5.2.3.20.1.3

从 $2 (x^{\frac{1}{5}}) + 2 (- 1)$ 中分解出因数 $2$ 。

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 + 10 x^{\frac{3}{5}} - 30 x^{\frac{2}{5}} + 30 x^{\frac{1}{5}} - 10 + \frac{5 {(2 (x^{\frac{1}{5}} - 1))}^{4}}{16} + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

解题步骤 1.5.2.3.20.2

对 $2 (x^{\frac{1}{5}} - 1)$ 运用乘积法则。

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 + 10 x^{\frac{3}{5}} - 30 x^{\frac{2}{5}} + 30 x^{\frac{1}{5}} - 10 + \frac{5 \cdot (2^{4} {(x^{\frac{1}{5}} - 1)}^{4})}{16} + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

解题步骤 1.5.2.3.20.3

对 $2$ 进行 $4$ 次方运算。

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 + 10 x^{\frac{3}{5}} - 30 x^{\frac{2}{5}} + 30 x^{\frac{1}{5}} - 10 + \frac{5 \cdot (16 {(x^{\frac{1}{5}} - 1)}^{4})}{16} + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

解题步骤 1.5.2.3.20.4

将 $5$ 乘以 $16$ 。

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 + 10 x^{\frac{3}{5}} - 30 x^{\frac{2}{5}} + 30 x^{\frac{1}{5}} - 10 + \frac{80 {(x^{\frac{1}{5}} - 1)}^{4}}{16} + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

解题步骤 1.5.2.3.21

从 $80 {(x^{\frac{1}{5}} - 1)}^{4}$ 中分解出因数 $16$ 。

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 + 10 x^{\frac{3}{5}} - 30 x^{\frac{2}{5}} + 30 x^{\frac{1}{5}} - 10 + \frac{16 (5 {(x^{\frac{1}{5}} - 1)}^{4})}{16} + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

解题步骤 1.5.2.3.22

约去公因数。

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解题步骤 1.5.2.3.22.1

从 $16$ 中分解出因数 $16$ 。

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 + 10 x^{\frac{3}{5}} - 30 x^{\frac{2}{5}} + 30 x^{\frac{1}{5}} - 10 + \frac{16 (5 {(x^{\frac{1}{5}} - 1)}^{4})}{16 (1)} + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

解题步骤 1.5.2.3.22.2

约去公因数。

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 + 10 x^{\frac{3}{5}} - 30 x^{\frac{2}{5}} + 30 x^{\frac{1}{5}} - 10 + \frac{16 (5 {(x^{\frac{1}{5}} - 1)}^{4})}{16 \cdot 1} + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

解题步骤 1.5.2.3.22.3

重写表达式。

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 + 10 x^{\frac{3}{5}} - 30 x^{\frac{2}{5}} + 30 x^{\frac{1}{5}} - 10 + \frac{5 {(x^{\frac{1}{5}} - 1)}^{4}}{1} + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

解题步骤 1.5.2.3.22.4

用 $5 {(x^{\frac{1}{5}} - 1)}^{4}$ 除以 $1$ 。

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 + 10 x^{\frac{3}{5}} - 30 x^{\frac{2}{5}} + 30 x^{\frac{1}{5}} - 10 + 5 {(x^{\frac{1}{5}} - 1)}^{4} + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

解题步骤 1.5.2.3.23

使用二项式定理。

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 + 10 x^{\frac{3}{5}} - 30 x^{\frac{2}{5}} + 30 x^{\frac{1}{5}} - 10 + 5 ({(x^{\frac{1}{5}})}^{4} + 4 {(x^{\frac{1}{5}})}^{3} \cdot - 1 + 6 {(x^{\frac{1}{5}})}^{2} {(- 1)}^{2} + 4 x^{\frac{1}{5}} {(- 1)}^{3} + {(- 1)}^{4}) + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

解题步骤 1.5.2.3.24

化简每一项。

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解题步骤 1.5.2.3.24.1

将 ${(x^{\frac{1}{5}})}^{4}$ 中的指数相乘。

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解题步骤 1.5.2.3.24.1.1

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

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 + 10 x^{\frac{3}{5}} - 30 x^{\frac{2}{5}} + 30 x^{\frac{1}{5}} - 10 + 5 (x^{\frac{1}{5} \cdot 4} + 4 {(x^{\frac{1}{5}})}^{3} \cdot - 1 + 6 {(x^{\frac{1}{5}})}^{2} {(- 1)}^{2} + 4 x^{\frac{1}{5}} {(- 1)}^{3} + {(- 1)}^{4}) + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

解题步骤 1.5.2.3.24.1.2

组合 $\frac{1}{5}$ 和 $4$ 。

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 + 10 x^{\frac{3}{5}} - 30 x^{\frac{2}{5}} + 30 x^{\frac{1}{5}} - 10 + 5 (x^{\frac{4}{5}} + 4 {(x^{\frac{1}{5}})}^{3} \cdot - 1 + 6 {(x^{\frac{1}{5}})}^{2} {(- 1)}^{2} + 4 x^{\frac{1}{5}} {(- 1)}^{3} + {(- 1)}^{4}) + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

解题步骤 1.5.2.3.24.2

将 ${(x^{\frac{1}{5}})}^{3}$ 中的指数相乘。

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解题步骤 1.5.2.3.24.2.1

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

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 + 10 x^{\frac{3}{5}} - 30 x^{\frac{2}{5}} + 30 x^{\frac{1}{5}} - 10 + 5 (x^{\frac{4}{5}} + 4 x^{\frac{1}{5} \cdot 3} \cdot - 1 + 6 {(x^{\frac{1}{5}})}^{2} {(- 1)}^{2} + 4 x^{\frac{1}{5}} {(- 1)}^{3} + {(- 1)}^{4}) + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

解题步骤 1.5.2.3.24.2.2

组合 $\frac{1}{5}$ 和 $3$ 。

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 + 10 x^{\frac{3}{5}} - 30 x^{\frac{2}{5}} + 30 x^{\frac{1}{5}} - 10 + 5 (x^{\frac{4}{5}} + 4 x^{\frac{3}{5}} \cdot - 1 + 6 {(x^{\frac{1}{5}})}^{2} {(- 1)}^{2} + 4 x^{\frac{1}{5}} {(- 1)}^{3} + {(- 1)}^{4}) + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

解题步骤 1.5.2.3.24.3

将 $- 1$ 乘以 $4$ 。

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 + 10 x^{\frac{3}{5}} - 30 x^{\frac{2}{5}} + 30 x^{\frac{1}{5}} - 10 + 5 (x^{\frac{4}{5}} - 4 x^{\frac{3}{5}} + 6 {(x^{\frac{1}{5}})}^{2} {(- 1)}^{2} + 4 x^{\frac{1}{5}} {(- 1)}^{3} + {(- 1)}^{4}) + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

解题步骤 1.5.2.3.24.4

将 ${(x^{\frac{1}{5}})}^{2}$ 中的指数相乘。

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解题步骤 1.5.2.3.24.4.1

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

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 + 10 x^{\frac{3}{5}} - 30 x^{\frac{2}{5}} + 30 x^{\frac{1}{5}} - 10 + 5 (x^{\frac{4}{5}} - 4 x^{\frac{3}{5}} + 6 x^{\frac{1}{5} \cdot 2} {(- 1)}^{2} + 4 x^{\frac{1}{5}} {(- 1)}^{3} + {(- 1)}^{4}) + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

解题步骤 1.5.2.3.24.4.2

组合 $\frac{1}{5}$ 和 $2$ 。

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 + 10 x^{\frac{3}{5}} - 30 x^{\frac{2}{5}} + 30 x^{\frac{1}{5}} - 10 + 5 (x^{\frac{4}{5}} - 4 x^{\frac{3}{5}} + 6 x^{\frac{2}{5}} {(- 1)}^{2} + 4 x^{\frac{1}{5}} {(- 1)}^{3} + {(- 1)}^{4}) + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

解题步骤 1.5.2.3.24.5

对 $- 1$ 进行 $2$ 次方运算。

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 + 10 x^{\frac{3}{5}} - 30 x^{\frac{2}{5}} + 30 x^{\frac{1}{5}} - 10 + 5 (x^{\frac{4}{5}} - 4 x^{\frac{3}{5}} + 6 x^{\frac{2}{5}} \cdot 1 + 4 x^{\frac{1}{5}} {(- 1)}^{3} + {(- 1)}^{4}) + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

解题步骤 1.5.2.3.24.6

将 $6$ 乘以 $1$ 。

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 + 10 x^{\frac{3}{5}} - 30 x^{\frac{2}{5}} + 30 x^{\frac{1}{5}} - 10 + 5 (x^{\frac{4}{5}} - 4 x^{\frac{3}{5}} + 6 x^{\frac{2}{5}} + 4 x^{\frac{1}{5}} {(- 1)}^{3} + {(- 1)}^{4}) + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

解题步骤 1.5.2.3.24.7

对 $- 1$ 进行 $3$ 次方运算。

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 + 10 x^{\frac{3}{5}} - 30 x^{\frac{2}{5}} + 30 x^{\frac{1}{5}} - 10 + 5 (x^{\frac{4}{5}} - 4 x^{\frac{3}{5}} + 6 x^{\frac{2}{5}} + 4 x^{\frac{1}{5}} \cdot - 1 + {(- 1)}^{4}) + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

解题步骤 1.5.2.3.24.8

将 $- 1$ 乘以 $4$ 。

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 + 10 x^{\frac{3}{5}} - 30 x^{\frac{2}{5}} + 30 x^{\frac{1}{5}} - 10 + 5 (x^{\frac{4}{5}} - 4 x^{\frac{3}{5}} + 6 x^{\frac{2}{5}} - 4 x^{\frac{1}{5}} + {(- 1)}^{4}) + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

解题步骤 1.5.2.3.24.9

对 $- 1$ 进行 $4$ 次方运算。

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 + 10 x^{\frac{3}{5}} - 30 x^{\frac{2}{5}} + 30 x^{\frac{1}{5}} - 10 + 5 (x^{\frac{4}{5}} - 4 x^{\frac{3}{5}} + 6 x^{\frac{2}{5}} - 4 x^{\frac{1}{5}} + 1) + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

解题步骤 1.5.2.3.25

运用分配律。

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 + 10 x^{\frac{3}{5}} - 30 x^{\frac{2}{5}} + 30 x^{\frac{1}{5}} - 10 + 5 x^{\frac{4}{5}} + 5 (- 4 x^{\frac{3}{5}}) + 5 (6 x^{\frac{2}{5}}) + 5 (- 4 x^{\frac{1}{5}}) + 5 \cdot 1 + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

解题步骤 1.5.2.3.26

化简。

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解题步骤 1.5.2.3.26.1

将 $- 4$ 乘以 $5$ 。

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 + 10 x^{\frac{3}{5}} - 30 x^{\frac{2}{5}} + 30 x^{\frac{1}{5}} - 10 + 5 x^{\frac{4}{5}} - 20 x^{\frac{3}{5}} + 5 (6 x^{\frac{2}{5}}) + 5 (- 4 x^{\frac{1}{5}}) + 5 \cdot 1 + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

解题步骤 1.5.2.3.26.2

将 $6$ 乘以 $5$ 。

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 + 10 x^{\frac{3}{5}} - 30 x^{\frac{2}{5}} + 30 x^{\frac{1}{5}} - 10 + 5 x^{\frac{4}{5}} - 20 x^{\frac{3}{5}} + 30 x^{\frac{2}{5}} + 5 (- 4 x^{\frac{1}{5}}) + 5 \cdot 1 + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

解题步骤 1.5.2.3.26.3

将 $- 4$ 乘以 $5$ 。

解题步骤 1.5.2.3.26.4

将 $5$ 乘以 $1$ 。

解题步骤 1.5.2.3.27

化简分子。

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解题步骤 1.5.2.3.27.1

从 $2 x^{\frac{1}{5}} - 2$ 中分解出因数 $2$ 。

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解题步骤 1.5.2.3.27.1.1

从 $2 x^{\frac{1}{5}}$ 中分解出因数 $2$ 。

解题步骤 1.5.2.3.27.1.2

从 $- 2$ 中分解出因数 $2$ 。

解题步骤 1.5.2.3.27.1.3

从 $2 (x^{\frac{1}{5}}) + 2 (- 1)$ 中分解出因数 $2$ 。

解题步骤 1.5.2.3.27.2

对 $2 (x^{\frac{1}{5}} - 1)$ 运用乘积法则。

解题步骤 1.5.2.3.27.3

对 $2$ 进行 $5$ 次方运算。

解题步骤 1.5.2.3.28

约去公因数。

解题步骤 1.5.2.3.29

用 ${(x^{\frac{1}{5}} - 1)}^{5}$ 除以 $1$ 。

解题步骤 1.5.2.3.30

使用二项式定理。

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 + 10 x^{\frac{3}{5}} - 30 x^{\frac{2}{5}} + 30 x^{\frac{1}{5}} - 10 + 5 x^{\frac{4}{5}} - 20 x^{\frac{3}{5}} + 30 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 5 + {(x^{\frac{1}{5}})}^{5} + 5 {(x^{\frac{1}{5}})}^{4} \cdot - 1 + 10 {(x^{\frac{1}{5}})}^{3} {(- 1)}^{2} + 10 {(x^{\frac{1}{5}})}^{2} {(- 1)}^{3} + 5 x^{\frac{1}{5}} {(- 1)}^{4} + {(- 1)}^{5}$

解题步骤 1.5.2.3.31

化简每一项。

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解题步骤 1.5.2.3.31.1

将 ${(x^{\frac{1}{5}})}^{5}$ 中的指数相乘。

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解题步骤 1.5.2.3.31.1.1

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

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 + 10 x^{\frac{3}{5}} - 30 x^{\frac{2}{5}} + 30 x^{\frac{1}{5}} - 10 + 5 x^{\frac{4}{5}} - 20 x^{\frac{3}{5}} + 30 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 5 + x^{\frac{1}{5} \cdot 5} + 5 {(x^{\frac{1}{5}})}^{4} \cdot - 1 + 10 {(x^{\frac{1}{5}})}^{3} {(- 1)}^{2} + 10 {(x^{\frac{1}{5}})}^{2} {(- 1)}^{3} + 5 x^{\frac{1}{5}} {(- 1)}^{4} + {(- 1)}^{5}$

解题步骤 1.5.2.3.31.1.2

约去 $5$ 的公因数。

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解题步骤 1.5.2.3.31.1.2.1

约去公因数。

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 + 10 x^{\frac{3}{5}} - 30 x^{\frac{2}{5}} + 30 x^{\frac{1}{5}} - 10 + 5 x^{\frac{4}{5}} - 20 x^{\frac{3}{5}} + 30 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 5 + x^{\frac{1}{5} \cdot 5} + 5 {(x^{\frac{1}{5}})}^{4} \cdot - 1 + 10 {(x^{\frac{1}{5}})}^{3} {(- 1)}^{2} + 10 {(x^{\frac{1}{5}})}^{2} {(- 1)}^{3} + 5 x^{\frac{1}{5}} {(- 1)}^{4} + {(- 1)}^{5}$

解题步骤 1.5.2.3.31.1.2.2

重写表达式。

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 + 10 x^{\frac{3}{5}} - 30 x^{\frac{2}{5}} + 30 x^{\frac{1}{5}} - 10 + 5 x^{\frac{4}{5}} - 20 x^{\frac{3}{5}} + 30 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 5 + x + 5 {(x^{\frac{1}{5}})}^{4} \cdot - 1 + 10 {(x^{\frac{1}{5}})}^{3} {(- 1)}^{2} + 10 {(x^{\frac{1}{5}})}^{2} {(- 1)}^{3} + 5 x^{\frac{1}{5}} {(- 1)}^{4} + {(- 1)}^{5}$

解题步骤 1.5.2.3.31.2

化简。

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 + 10 x^{\frac{3}{5}} - 30 x^{\frac{2}{5}} + 30 x^{\frac{1}{5}} - 10 + 5 x^{\frac{4}{5}} - 20 x^{\frac{3}{5}} + 30 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 5 + x + 5 {(x^{\frac{1}{5}})}^{4} \cdot - 1 + 10 {(x^{\frac{1}{5}})}^{3} {(- 1)}^{2} + 10 {(x^{\frac{1}{5}})}^{2} {(- 1)}^{3} + 5 x^{\frac{1}{5}} {(- 1)}^{4} + {(- 1)}^{5}$

解题步骤 1.5.2.3.31.3

将 ${(x^{\frac{1}{5}})}^{4}$ 中的指数相乘。

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解题步骤 1.5.2.3.31.3.1

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

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 + 10 x^{\frac{3}{5}} - 30 x^{\frac{2}{5}} + 30 x^{\frac{1}{5}} - 10 + 5 x^{\frac{4}{5}} - 20 x^{\frac{3}{5}} + 30 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 5 + x + 5 x^{\frac{1}{5} \cdot 4} \cdot - 1 + 10 {(x^{\frac{1}{5}})}^{3} {(- 1)}^{2} + 10 {(x^{\frac{1}{5}})}^{2} {(- 1)}^{3} + 5 x^{\frac{1}{5}} {(- 1)}^{4} + {(- 1)}^{5}$

解题步骤 1.5.2.3.31.3.2

组合 $\frac{1}{5}$ 和 $4$ 。

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 + 10 x^{\frac{3}{5}} - 30 x^{\frac{2}{5}} + 30 x^{\frac{1}{5}} - 10 + 5 x^{\frac{4}{5}} - 20 x^{\frac{3}{5}} + 30 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 5 + x + 5 x^{\frac{4}{5}} \cdot - 1 + 10 {(x^{\frac{1}{5}})}^{3} {(- 1)}^{2} + 10 {(x^{\frac{1}{5}})}^{2} {(- 1)}^{3} + 5 x^{\frac{1}{5}} {(- 1)}^{4} + {(- 1)}^{5}$

解题步骤 1.5.2.3.31.4

将 $- 1$ 乘以 $5$ 。

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 + 10 x^{\frac{3}{5}} - 30 x^{\frac{2}{5}} + 30 x^{\frac{1}{5}} - 10 + 5 x^{\frac{4}{5}} - 20 x^{\frac{3}{5}} + 30 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 5 + x - 5 x^{\frac{4}{5}} + 10 {(x^{\frac{1}{5}})}^{3} {(- 1)}^{2} + 10 {(x^{\frac{1}{5}})}^{2} {(- 1)}^{3} + 5 x^{\frac{1}{5}} {(- 1)}^{4} + {(- 1)}^{5}$

解题步骤 1.5.2.3.31.5

将 ${(x^{\frac{1}{5}})}^{3}$ 中的指数相乘。

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解题步骤 1.5.2.3.31.5.1

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

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 + 10 x^{\frac{3}{5}} - 30 x^{\frac{2}{5}} + 30 x^{\frac{1}{5}} - 10 + 5 x^{\frac{4}{5}} - 20 x^{\frac{3}{5}} + 30 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 5 + x - 5 x^{\frac{4}{5}} + 10 x^{\frac{1}{5} \cdot 3} {(- 1)}^{2} + 10 {(x^{\frac{1}{5}})}^{2} {(- 1)}^{3} + 5 x^{\frac{1}{5}} {(- 1)}^{4} + {(- 1)}^{5}$

解题步骤 1.5.2.3.31.5.2

组合 $\frac{1}{5}$ 和 $3$ 。

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 + 10 x^{\frac{3}{5}} - 30 x^{\frac{2}{5}} + 30 x^{\frac{1}{5}} - 10 + 5 x^{\frac{4}{5}} - 20 x^{\frac{3}{5}} + 30 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 5 + x - 5 x^{\frac{4}{5}} + 10 x^{\frac{3}{5}} {(- 1)}^{2} + 10 {(x^{\frac{1}{5}})}^{2} {(- 1)}^{3} + 5 x^{\frac{1}{5}} {(- 1)}^{4} + {(- 1)}^{5}$

解题步骤 1.5.2.3.31.6

对 $- 1$ 进行 $2$ 次方运算。

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 + 10 x^{\frac{3}{5}} - 30 x^{\frac{2}{5}} + 30 x^{\frac{1}{5}} - 10 + 5 x^{\frac{4}{5}} - 20 x^{\frac{3}{5}} + 30 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 5 + x - 5 x^{\frac{4}{5}} + 10 x^{\frac{3}{5}} \cdot 1 + 10 {(x^{\frac{1}{5}})}^{2} {(- 1)}^{3} + 5 x^{\frac{1}{5}} {(- 1)}^{4} + {(- 1)}^{5}$

解题步骤 1.5.2.3.31.7

将 $10$ 乘以 $1$ 。

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 + 10 x^{\frac{3}{5}} - 30 x^{\frac{2}{5}} + 30 x^{\frac{1}{5}} - 10 + 5 x^{\frac{4}{5}} - 20 x^{\frac{3}{5}} + 30 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 5 + x - 5 x^{\frac{4}{5}} + 10 x^{\frac{3}{5}} + 10 {(x^{\frac{1}{5}})}^{2} {(- 1)}^{3} + 5 x^{\frac{1}{5}} {(- 1)}^{4} + {(- 1)}^{5}$

解题步骤 1.5.2.3.31.8

将 ${(x^{\frac{1}{5}})}^{2}$ 中的指数相乘。

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解题步骤 1.5.2.3.31.8.1

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

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 + 10 x^{\frac{3}{5}} - 30 x^{\frac{2}{5}} + 30 x^{\frac{1}{5}} - 10 + 5 x^{\frac{4}{5}} - 20 x^{\frac{3}{5}} + 30 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 5 + x - 5 x^{\frac{4}{5}} + 10 x^{\frac{3}{5}} + 10 x^{\frac{1}{5} \cdot 2} {(- 1)}^{3} + 5 x^{\frac{1}{5}} {(- 1)}^{4} + {(- 1)}^{5}$

解题步骤 1.5.2.3.31.8.2

组合 $\frac{1}{5}$ 和 $2$ 。

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 + 10 x^{\frac{3}{5}} - 30 x^{\frac{2}{5}} + 30 x^{\frac{1}{5}} - 10 + 5 x^{\frac{4}{5}} - 20 x^{\frac{3}{5}} + 30 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 5 + x - 5 x^{\frac{4}{5}} + 10 x^{\frac{3}{5}} + 10 x^{\frac{2}{5}} {(- 1)}^{3} + 5 x^{\frac{1}{5}} {(- 1)}^{4} + {(- 1)}^{5}$

解题步骤 1.5.2.3.31.9

对 $- 1$ 进行 $3$ 次方运算。

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 + 10 x^{\frac{3}{5}} - 30 x^{\frac{2}{5}} + 30 x^{\frac{1}{5}} - 10 + 5 x^{\frac{4}{5}} - 20 x^{\frac{3}{5}} + 30 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 5 + x - 5 x^{\frac{4}{5}} + 10 x^{\frac{3}{5}} + 10 x^{\frac{2}{5}} \cdot - 1 + 5 x^{\frac{1}{5}} {(- 1)}^{4} + {(- 1)}^{5}$

解题步骤 1.5.2.3.31.10

将 $- 1$ 乘以 $10$ 。

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 + 10 x^{\frac{3}{5}} - 30 x^{\frac{2}{5}} + 30 x^{\frac{1}{5}} - 10 + 5 x^{\frac{4}{5}} - 20 x^{\frac{3}{5}} + 30 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 5 + x - 5 x^{\frac{4}{5}} + 10 x^{\frac{3}{5}} - 10 x^{\frac{2}{5}} + 5 x^{\frac{1}{5}} {(- 1)}^{4} + {(- 1)}^{5}$

解题步骤 1.5.2.3.31.11

对 $- 1$ 进行 $4$ 次方运算。

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 + 10 x^{\frac{3}{5}} - 30 x^{\frac{2}{5}} + 30 x^{\frac{1}{5}} - 10 + 5 x^{\frac{4}{5}} - 20 x^{\frac{3}{5}} + 30 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 5 + x - 5 x^{\frac{4}{5}} + 10 x^{\frac{3}{5}} - 10 x^{\frac{2}{5}} + 5 x^{\frac{1}{5}} \cdot 1 + {(- 1)}^{5}$

解题步骤 1.5.2.3.31.12

将 $5$ 乘以 $1$ 。

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 + 10 x^{\frac{3}{5}} - 30 x^{\frac{2}{5}} + 30 x^{\frac{1}{5}} - 10 + 5 x^{\frac{4}{5}} - 20 x^{\frac{3}{5}} + 30 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 5 + x - 5 x^{\frac{4}{5}} + 10 x^{\frac{3}{5}} - 10 x^{\frac{2}{5}} + 5 x^{\frac{1}{5}} + {(- 1)}^{5}$

解题步骤 1.5.2.3.31.13

对 $- 1$ 进行 $5$ 次方运算。

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 + 10 x^{\frac{3}{5}} - 30 x^{\frac{2}{5}} + 30 x^{\frac{1}{5}} - 10 + 5 x^{\frac{4}{5}} - 20 x^{\frac{3}{5}} + 30 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 5 + x - 5 x^{\frac{4}{5}} + 10 x^{\frac{3}{5}} - 10 x^{\frac{2}{5}} + 5 x^{\frac{1}{5}} - 1$

解题步骤 1.5.2.4

通过加上各项进行化简。

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解题步骤 1.5.2.4.1

合并 $1 + 5 x^{\frac{1}{5}} - 5 + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 + 10 x^{\frac{3}{5}} - 30 x^{\frac{2}{5}} + 30 x^{\frac{1}{5}} - 10 + 5 x^{\frac{4}{5}} - 20 x^{\frac{3}{5}} + 30 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 5 + x - 5 x^{\frac{4}{5}} + 10 x^{\frac{3}{5}} - 10 x^{\frac{2}{5}} + 5 x^{\frac{1}{5}} - 1$ 中相反的项。

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解题步骤 1.5.2.4.1.1

从 $10$ 中减去 $10$ 。

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 x^{\frac{3}{5}} - 30 x^{\frac{2}{5}} + 30 x^{\frac{1}{5}} + 0 + 5 x^{\frac{4}{5}} - 20 x^{\frac{3}{5}} + 30 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 5 + x - 5 x^{\frac{4}{5}} + 10 x^{\frac{3}{5}} - 10 x^{\frac{2}{5}} + 5 x^{\frac{1}{5}} - 1$

解题步骤 1.5.2.4.1.2

将 $1 + 5 x^{\frac{1}{5}} - 5 + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 x^{\frac{3}{5}} - 30 x^{\frac{2}{5}} + 30 x^{\frac{1}{5}}$ 和 $0$ 相加。

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 x^{\frac{3}{5}} - 30 x^{\frac{2}{5}} + 30 x^{\frac{1}{5}} + 5 x^{\frac{4}{5}} - 20 x^{\frac{3}{5}} + 30 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 5 + x - 5 x^{\frac{4}{5}} + 10 x^{\frac{3}{5}} - 10 x^{\frac{2}{5}} + 5 x^{\frac{1}{5}} - 1$

解题步骤 1.5.2.4.1.3

将 $- 30 x^{\frac{2}{5}}$ 和 $30 x^{\frac{2}{5}}$ 相加。

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 x^{\frac{3}{5}} + 30 x^{\frac{1}{5}} + 5 x^{\frac{4}{5}} - 20 x^{\frac{3}{5}} + 0 - 20 x^{\frac{1}{5}} + 5 + x - 5 x^{\frac{4}{5}} + 10 x^{\frac{3}{5}} - 10 x^{\frac{2}{5}} + 5 x^{\frac{1}{5}} - 1$

解题步骤 1.5.2.4.1.4

将 $1 + 5 x^{\frac{1}{5}} - 5 + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 x^{\frac{3}{5}} + 30 x^{\frac{1}{5}} + 5 x^{\frac{4}{5}} - 20 x^{\frac{3}{5}}$ 和 $0$ 相加。

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 x^{\frac{3}{5}} + 30 x^{\frac{1}{5}} + 5 x^{\frac{4}{5}} - 20 x^{\frac{3}{5}} - 20 x^{\frac{1}{5}} + 5 + x - 5 x^{\frac{4}{5}} + 10 x^{\frac{3}{5}} - 10 x^{\frac{2}{5}} + 5 x^{\frac{1}{5}} - 1$

解题步骤 1.5.2.4.1.5

将 $- 5$ 和 $5$ 相加。

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 x^{\frac{3}{5}} + 30 x^{\frac{1}{5}} + 5 x^{\frac{4}{5}} - 20 x^{\frac{3}{5}} - 20 x^{\frac{1}{5}} + 0 + x - 5 x^{\frac{4}{5}} + 10 x^{\frac{3}{5}} - 10 x^{\frac{2}{5}} + 5 x^{\frac{1}{5}} - 1$

解题步骤 1.5.2.4.1.6

将 $1 + 5 x^{\frac{1}{5}} + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 x^{\frac{3}{5}} + 30 x^{\frac{1}{5}} + 5 x^{\frac{4}{5}} - 20 x^{\frac{3}{5}} - 20 x^{\frac{1}{5}}$ 和 $0$ 相加。

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 x^{\frac{3}{5}} + 30 x^{\frac{1}{5}} + 5 x^{\frac{4}{5}} - 20 x^{\frac{3}{5}} - 20 x^{\frac{1}{5}} + x - 5 x^{\frac{4}{5}} + 10 x^{\frac{3}{5}} - 10 x^{\frac{2}{5}} + 5 x^{\frac{1}{5}} - 1$

解题步骤 1.5.2.4.1.7

从 $5 x^{\frac{4}{5}}$ 中减去 $5 x^{\frac{4}{5}}$ 。

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 x^{\frac{3}{5}} + 30 x^{\frac{1}{5}} - 20 x^{\frac{3}{5}} - 20 x^{\frac{1}{5}} + x + 0 + 10 x^{\frac{3}{5}} - 10 x^{\frac{2}{5}} + 5 x^{\frac{1}{5}} - 1$

解题步骤 1.5.2.4.1.8

将 $1 + 5 x^{\frac{1}{5}} + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 x^{\frac{3}{5}} + 30 x^{\frac{1}{5}} - 20 x^{\frac{3}{5}} - 20 x^{\frac{1}{5}} + x$ 和 $0$ 相加。

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 x^{\frac{3}{5}} + 30 x^{\frac{1}{5}} - 20 x^{\frac{3}{5}} - 20 x^{\frac{1}{5}} + x + 10 x^{\frac{3}{5}} - 10 x^{\frac{2}{5}} + 5 x^{\frac{1}{5}} - 1$

解题步骤 1.5.2.4.1.9

从 $10 x^{\frac{2}{5}}$ 中减去 $10 x^{\frac{2}{5}}$ 。

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 20 x^{\frac{1}{5}} + 10 x^{\frac{3}{5}} + 30 x^{\frac{1}{5}} - 20 x^{\frac{3}{5}} - 20 x^{\frac{1}{5}} + x + 10 x^{\frac{3}{5}} + 0 + 5 x^{\frac{1}{5}} - 1$

解题步骤 1.5.2.4.1.10

将 $1 + 5 x^{\frac{1}{5}} - 20 x^{\frac{1}{5}} + 10 x^{\frac{3}{5}} + 30 x^{\frac{1}{5}} - 20 x^{\frac{3}{5}} - 20 x^{\frac{1}{5}} + x + 10 x^{\frac{3}{5}}$ 和 $0$ 相加。

解题步骤 1.5.2.4.1.11

从 $1$ 中减去 $1$ 。

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 5 x^{\frac{1}{5}} - 20 x^{\frac{1}{5}} + 10 x^{\frac{3}{5}} + 30 x^{\frac{1}{5}} - 20 x^{\frac{3}{5}} - 20 x^{\frac{1}{5}} + x + 10 x^{\frac{3}{5}} + 5 x^{\frac{1}{5}} + 0$

解题步骤 1.5.2.4.1.12

将 $5 x^{\frac{1}{5}} - 20 x^{\frac{1}{5}} + 10 x^{\frac{3}{5}} + 30 x^{\frac{1}{5}} - 20 x^{\frac{3}{5}} - 20 x^{\frac{1}{5}} + x + 10 x^{\frac{3}{5}} + 5 x^{\frac{1}{5}}$ 和 $0$ 相加。

解题步骤 1.5.2.4.2

从 $5 x^{\frac{1}{5}}$ 中减去 $20 x^{\frac{1}{5}}$ 。

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = - 15 x^{\frac{1}{5}} + 10 x^{\frac{3}{5}} + 30 x^{\frac{1}{5}} - 20 x^{\frac{3}{5}} - 20 x^{\frac{1}{5}} + x + 10 x^{\frac{3}{5}} + 5 x^{\frac{1}{5}}$

解题步骤 1.5.2.4.3

将 $- 15 x^{\frac{1}{5}}$ 和 $30 x^{\frac{1}{5}}$ 相加。

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 15 x^{\frac{1}{5}} + 10 x^{\frac{3}{5}} - 20 x^{\frac{3}{5}} - 20 x^{\frac{1}{5}} + x + 10 x^{\frac{3}{5}} + 5 x^{\frac{1}{5}}$

解题步骤 1.5.2.4.4

从 $15 x^{\frac{1}{5}}$ 中减去 $20 x^{\frac{1}{5}}$ 。

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = - 5 x^{\frac{1}{5}} + 10 x^{\frac{3}{5}} - 20 x^{\frac{3}{5}} + x + 10 x^{\frac{3}{5}} + 5 x^{\frac{1}{5}}$

解题步骤 1.5.2.4.5

合并 $- 5 x^{\frac{1}{5}} + 10 x^{\frac{3}{5}} - 20 x^{\frac{3}{5}} + x + 10 x^{\frac{3}{5}} + 5 x^{\frac{1}{5}}$ 中相反的项。

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解题步骤 1.5.2.4.5.1

将 $- 5 x^{\frac{1}{5}}$ 和 $5 x^{\frac{1}{5}}$ 相加。

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 10 x^{\frac{3}{5}} - 20 x^{\frac{3}{5}} + x + 10 x^{\frac{3}{5}} + 0$

解题步骤 1.5.2.4.5.2

将 $10 x^{\frac{3}{5}} - 20 x^{\frac{3}{5}} + x + 10 x^{\frac{3}{5}}$ 和 $0$ 相加。

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 10 x^{\frac{3}{5}} - 20 x^{\frac{3}{5}} + x + 10 x^{\frac{3}{5}}$

解题步骤 1.5.2.4.6

从 $10 x^{\frac{3}{5}}$ 中减去 $20 x^{\frac{3}{5}}$ 。

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = - 10 x^{\frac{3}{5}} + x + 10 x^{\frac{3}{5}}$

解题步骤 1.5.2.4.7

合并 $- 10 x^{\frac{3}{5}} + x + 10 x^{\frac{3}{5}}$ 中相反的项。

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解题步骤 1.5.2.4.7.1

将 $- 10 x^{\frac{3}{5}}$ 和 $10 x^{\frac{3}{5}}$ 相加。

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

解题步骤 1.5.2.4.7.2

将 $x$ 和 $0$ 相加。

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

解题步骤 1.5.3

计算 $f (f^{- 1} (x))$ 。

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解题步骤 1.5.3.1

建立复合结果函数。

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

解题步骤 1.5.3.2

通过将 $f^{- 1}$ 的值代入 $f$ 来计算 $f (1 + \frac{5 x}{2} + \frac{5 x^{2}}{2} + \frac{5 x^{3}}{4} + \frac{5 x^{4}}{16} + \frac{x^{5}}{32})$ 。

$f (1 + \frac{5 x}{2} + \frac{5 x^{2}}{2} + \frac{5 x^{3}}{4} + \frac{5 x^{4}}{16} + \frac{x^{5}}{32}) = 2 {(1 + \frac{5 x}{2} + \frac{5 x^{2}}{2} + \frac{5 x^{3}}{4} + \frac{5 x^{4}}{16} + \frac{x^{5}}{32})}^{\frac{1}{5}} - 2$

解题步骤 1.5.3.3

移动 $1$ 。

$f (1 + \frac{5 x}{2} + \frac{5 x^{2}}{2} + \frac{5 x^{3}}{4} + \frac{5 x^{4}}{16} + \frac{x^{5}}{32}) = 2 {(\frac{5 x}{2} + \frac{5 x^{2}}{2} + \frac{5 x^{3}}{4} + \frac{5 x^{4}}{16} + \frac{x^{5}}{32} + 1)}^{\frac{1}{5}} - 2$

解题步骤 1.5.3.4

移动 $\frac{5 x}{2}$ 。

$f (1 + \frac{5 x}{2} + \frac{5 x^{2}}{2} + \frac{5 x^{3}}{4} + \frac{5 x^{4}}{16} + \frac{x^{5}}{32}) = 2 {(\frac{5 x^{2}}{2} + \frac{5 x^{3}}{4} + \frac{5 x^{4}}{16} + \frac{x^{5}}{32} + \frac{5 x}{2} + 1)}^{\frac{1}{5}} - 2$

解题步骤 1.5.3.5

移动 $\frac{5 x^{2}}{2}$ 。

$f (1 + \frac{5 x}{2} + \frac{5 x^{2}}{2} + \frac{5 x^{3}}{4} + \frac{5 x^{4}}{16} + \frac{x^{5}}{32}) = 2 {(\frac{5 x^{3}}{4} + \frac{5 x^{4}}{16} + \frac{x^{5}}{32} + \frac{5 x^{2}}{2} + \frac{5 x}{2} + 1)}^{\frac{1}{5}} - 2$

解题步骤 1.5.3.6

移动 $\frac{5 x^{3}}{4}$ 。

$f (1 + \frac{5 x}{2} + \frac{5 x^{2}}{2} + \frac{5 x^{3}}{4} + \frac{5 x^{4}}{16} + \frac{x^{5}}{32}) = 2 {(\frac{5 x^{4}}{16} + \frac{x^{5}}{32} + \frac{5 x^{3}}{4} + \frac{5 x^{2}}{2} + \frac{5 x}{2} + 1)}^{\frac{1}{5}} - 2$

解题步骤 1.5.3.7

将 $\frac{5 x^{4}}{16}$ 和 $\frac{x^{5}}{32}$ 重新排序。

$f (1 + \frac{5 x}{2} + \frac{5 x^{2}}{2} + \frac{5 x^{3}}{4} + \frac{5 x^{4}}{16} + \frac{x^{5}}{32}) = 2 {(\frac{x^{5}}{32} + \frac{5 x^{4}}{16} + \frac{5 x^{3}}{4} + \frac{5 x^{2}}{2} + \frac{5 x}{2} + 1)}^{\frac{1}{5}} - 2$

解题步骤 1.5.4

由于 $f^{- 1} (f (x)) = x$ 和 $f (f^{- 1} (x)) = x$ ，因此 $f^{- 1} (x) = 1 + \frac{5 x}{2} + \frac{5 x^{2}}{2} + \frac{5 x^{3}}{4} + \frac{5 x^{4}}{16} + \frac{x^{5}}{32}$ 为 $f (x) = 2 x^{\frac{1}{5}} - 2$ 的反函数。

$f^{- 1} (x) = 1 + \frac{5 x}{2} + \frac{5 x^{2}}{2} + \frac{5 x^{3}}{4} + \frac{5 x^{4}}{16} + \frac{x^{5}}{32}$

解题步骤 2

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解题步骤 2.1

移动 $1$ 。

$\frac{5 x}{2} + \frac{5 x^{2}}{2} + \frac{5 x^{3}}{4} + \frac{5 x^{4}}{16} + \frac{x^{5}}{32} + 1$

解题步骤 2.2

移动 $\frac{5 x}{2}$ 。

$\frac{5 x^{2}}{2} + \frac{5 x^{3}}{4} + \frac{5 x^{4}}{16} + \frac{x^{5}}{32} + \frac{5 x}{2} + 1$

解题步骤 2.3

移动 $\frac{5 x^{2}}{2}$ 。

$\frac{5 x^{3}}{4} + \frac{5 x^{4}}{16} + \frac{x^{5}}{32} + \frac{5 x^{2}}{2} + \frac{5 x}{2} + 1$

解题步骤 2.4

移动 $\frac{5 x^{3}}{4}$ 。

$\frac{5 x^{4}}{16} + \frac{x^{5}}{32} + \frac{5 x^{3}}{4} + \frac{5 x^{2}}{2} + \frac{5 x}{2} + 1$

解题步骤 2.5

将 $\frac{5 x^{4}}{16}$ 和 $\frac{x^{5}}{32}$ 重新排序。

$\frac{x^{5}}{32} + \frac{5 x^{4}}{16} + \frac{5 x^{3}}{4} + \frac{5 x^{2}}{2} + \frac{5 x}{2} + 1$

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