初级微积分示例

$f (x) = 7 (x^{\frac{1}{5}} - 10)$ ; find $f^{- 1} (x)$

解题步骤 1

将 $f (x) = 7 (x^{\frac{1}{5}} - 10)$ 写为等式。

$y = 7 (x^{\frac{1}{5}} - 10)$

解题步骤 2

交换变量。

$x = 7 (y^{\frac{1}{5}} - 10)$

解题步骤 3

求解 $y$ 。

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解题步骤 3.1

将方程重写为 $7 (y^{\frac{1}{5}} - 10) = x$ 。

$7 (y^{\frac{1}{5}} - 10) = x$

解题步骤 3.2

将 $7 (y^{\frac{1}{5}} - 10) = x$ 中的每一项除以 $7$ 并化简。

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解题步骤 3.2.1

将 $7 (y^{\frac{1}{5}} - 10) = x$ 中的每一项都除以 $7$ 。

$\frac{7 (y^{\frac{1}{5}} - 10)}{7} = \frac{x}{7}$

解题步骤 3.2.2

化简左边。

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解题步骤 3.2.2.1

约去公因数。

$\frac{7 (y^{\frac{1}{5}} - 10)}{7} = \frac{x}{7}$

解题步骤 3.2.2.2

用 $y^{\frac{1}{5}} - 10$ 除以 $1$ 。

$y^{\frac{1}{5}} - 10 = \frac{x}{7}$

解题步骤 3.3

将所有不包含 $y$ 的项移到等式右边。

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解题步骤 3.3.1

在等式两边都加上 $10$ 。

$y^{\frac{1}{5}} - \frac{x}{7} = 10$

解题步骤 3.3.2

在等式两边都加上 $\frac{x}{7}$ 。

$y^{\frac{1}{5}} = 10 + \frac{x}{7}$

解题步骤 3.4

将方程两边同时进行 $5$ 次方运算以消去左边的分数指数。

${(y^{\frac{1}{5}})}^{5} = {(10 + \frac{x}{7})}^{5}$

解题步骤 3.5

化简指数。

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解题步骤 3.5.1

化简左边。

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解题步骤 3.5.1.1

化简 ${(y^{\frac{1}{5}})}^{5}$ 。

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解题步骤 3.5.1.1.1

将 ${(y^{\frac{1}{5}})}^{5}$ 中的指数相乘。

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解题步骤 3.5.1.1.1.1

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

$y^{\frac{1}{5} \cdot 5} = {(10 + \frac{x}{7})}^{5}$

解题步骤 3.5.1.1.1.2

约去 $5$ 的公因数。

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解题步骤 3.5.1.1.1.2.1

约去公因数。

$y^{\frac{1}{5} \cdot 5} = {(10 + \frac{x}{7})}^{5}$

解题步骤 3.5.1.1.1.2.2

重写表达式。

$y^{1} = {(10 + \frac{x}{7})}^{5}$

解题步骤 3.5.1.1.2

化简。

$y = {(10 + \frac{x}{7})}^{5}$

解题步骤 3.5.2

化简右边。

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解题步骤 3.5.2.1

化简 ${(10 + \frac{x}{7})}^{5}$ 。

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解题步骤 3.5.2.1.1

使用二项式定理。

$y = 10^{5} + 5 \cdot 10^{4} \frac{x}{7} + 10 \cdot 10^{3} {(\frac{x}{7})}^{2} + 10 \cdot 10^{2} {(\frac{x}{7})}^{3} + 5 \cdot 10 {(\frac{x}{7})}^{4} + {(\frac{x}{7})}^{5}$

解题步骤 3.5.2.1.2

化简每一项。

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解题步骤 3.5.2.1.2.1

对 $10$ 进行 $5$ 次方运算。

$y = 100000 + 5 \cdot 10^{4} \frac{x}{7} + 10 \cdot 10^{3} {(\frac{x}{7})}^{2} + 10 \cdot 10^{2} {(\frac{x}{7})}^{3} + 5 \cdot 10 {(\frac{x}{7})}^{4} + {(\frac{x}{7})}^{5}$

解题步骤 3.5.2.1.2.2

乘以 $5 \cdot 10^{4} \frac{x}{7}$ 。

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解题步骤 3.5.2.1.2.2.1

组合 $5$ 和 $\frac{x}{7}$ 。

$y = 100000 + 10^{4} \frac{5 x}{7} + 10 \cdot 10^{3} {(\frac{x}{7})}^{2} + 10 \cdot 10^{2} {(\frac{x}{7})}^{3} + 5 \cdot 10 {(\frac{x}{7})}^{4} + {(\frac{x}{7})}^{5}$

解题步骤 3.5.2.1.2.2.2

组合 $10^{4}$ 和 $\frac{5 x}{7}$ 。

$y = 100000 + \frac{10^{4} (5 x)}{7} + 10 \cdot 10^{3} {(\frac{x}{7})}^{2} + 10 \cdot 10^{2} {(\frac{x}{7})}^{3} + 5 \cdot 10 {(\frac{x}{7})}^{4} + {(\frac{x}{7})}^{5}$

解题步骤 3.5.2.1.2.3

化简分子。

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解题步骤 3.5.2.1.2.3.1

对 $10$ 进行 $4$ 次方运算。

$y = 100000 + \frac{10000 \cdot 5 x}{7} + 10 \cdot 10^{3} {(\frac{x}{7})}^{2} + 10 \cdot 10^{2} {(\frac{x}{7})}^{3} + 5 \cdot 10 {(\frac{x}{7})}^{4} + {(\frac{x}{7})}^{5}$

解题步骤 3.5.2.1.2.3.2

将 $10000$ 乘以 $5$ 。

$y = 100000 + \frac{50000 x}{7} + 10 \cdot 10^{3} {(\frac{x}{7})}^{2} + 10 \cdot 10^{2} {(\frac{x}{7})}^{3} + 5 \cdot 10 {(\frac{x}{7})}^{4} + {(\frac{x}{7})}^{5}$

解题步骤 3.5.2.1.2.4

通过指数相加将 $10$ 乘以 $10^{3}$ 。

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解题步骤 3.5.2.1.2.4.1

将 $10$ 乘以 $10^{3}$ 。

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解题步骤 3.5.2.1.2.4.1.1

对 $10$ 进行 $1$ 次方运算。

$y = 100000 + \frac{50000 x}{7} + 10^{1} \cdot 10^{3} {(\frac{x}{7})}^{2} + 10 \cdot 10^{2} {(\frac{x}{7})}^{3} + 5 \cdot 10 {(\frac{x}{7})}^{4} + {(\frac{x}{7})}^{5}$

解题步骤 3.5.2.1.2.4.1.2

使用幂法则 $a^{m} a^{n} = a^{m + n}$ 合并指数。

$y = 100000 + \frac{50000 x}{7} + 10^{1 + 3} {(\frac{x}{7})}^{2} + 10 \cdot 10^{2} {(\frac{x}{7})}^{3} + 5 \cdot 10 {(\frac{x}{7})}^{4} + {(\frac{x}{7})}^{5}$

解题步骤 3.5.2.1.2.4.2

将 $1$ 和 $3$ 相加。

$y = 100000 + \frac{50000 x}{7} + 10^{4} {(\frac{x}{7})}^{2} + 10 \cdot 10^{2} {(\frac{x}{7})}^{3} + 5 \cdot 10 {(\frac{x}{7})}^{4} + {(\frac{x}{7})}^{5}$

解题步骤 3.5.2.1.2.5

对 $10$ 进行 $4$ 次方运算。

$y = 100000 + \frac{50000 x}{7} + 10000 {(\frac{x}{7})}^{2} + 10 \cdot 10^{2} {(\frac{x}{7})}^{3} + 5 \cdot 10 {(\frac{x}{7})}^{4} + {(\frac{x}{7})}^{5}$

解题步骤 3.5.2.1.2.6

对 $\frac{x}{7}$ 运用乘积法则。

$y = 100000 + \frac{50000 x}{7} + 10000 \frac{x^{2}}{7^{2}} + 10 \cdot 10^{2} {(\frac{x}{7})}^{3} + 5 \cdot 10 {(\frac{x}{7})}^{4} + {(\frac{x}{7})}^{5}$

解题步骤 3.5.2.1.2.7

对 $7$ 进行 $2$ 次方运算。

$y = 100000 + \frac{50000 x}{7} + 10000 \frac{x^{2}}{49} + 10 \cdot 10^{2} {(\frac{x}{7})}^{3} + 5 \cdot 10 {(\frac{x}{7})}^{4} + {(\frac{x}{7})}^{5}$

解题步骤 3.5.2.1.2.8

组合 $10000$ 和 $\frac{x^{2}}{49}$ 。

$y = 100000 + \frac{50000 x}{7} + \frac{10000 x^{2}}{49} + 10 \cdot 10^{2} {(\frac{x}{7})}^{3} + 5 \cdot 10 {(\frac{x}{7})}^{4} + {(\frac{x}{7})}^{5}$

解题步骤 3.5.2.1.2.9

通过指数相加将 $10$ 乘以 $10^{2}$ 。

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解题步骤 3.5.2.1.2.9.1

将 $10$ 乘以 $10^{2}$ 。

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解题步骤 3.5.2.1.2.9.1.1

对 $10$ 进行 $1$ 次方运算。

$y = 100000 + \frac{50000 x}{7} + \frac{10000 x^{2}}{49} + 10^{1} \cdot 10^{2} {(\frac{x}{7})}^{3} + 5 \cdot 10 {(\frac{x}{7})}^{4} + {(\frac{x}{7})}^{5}$

解题步骤 3.5.2.1.2.9.1.2

使用幂法则 $a^{m} a^{n} = a^{m + n}$ 合并指数。

$y = 100000 + \frac{50000 x}{7} + \frac{10000 x^{2}}{49} + 10^{1 + 2} {(\frac{x}{7})}^{3} + 5 \cdot 10 {(\frac{x}{7})}^{4} + {(\frac{x}{7})}^{5}$

解题步骤 3.5.2.1.2.9.2

将 $1$ 和 $2$ 相加。

$y = 100000 + \frac{50000 x}{7} + \frac{10000 x^{2}}{49} + 10^{3} {(\frac{x}{7})}^{3} + 5 \cdot 10 {(\frac{x}{7})}^{4} + {(\frac{x}{7})}^{5}$

解题步骤 3.5.2.1.2.10

对 $10$ 进行 $3$ 次方运算。

$y = 100000 + \frac{50000 x}{7} + \frac{10000 x^{2}}{49} + 1000 {(\frac{x}{7})}^{3} + 5 \cdot 10 {(\frac{x}{7})}^{4} + {(\frac{x}{7})}^{5}$

解题步骤 3.5.2.1.2.11

对 $\frac{x}{7}$ 运用乘积法则。

$y = 100000 + \frac{50000 x}{7} + \frac{10000 x^{2}}{49} + 1000 \frac{x^{3}}{7^{3}} + 5 \cdot 10 {(\frac{x}{7})}^{4} + {(\frac{x}{7})}^{5}$

解题步骤 3.5.2.1.2.12

对 $7$ 进行 $3$ 次方运算。

$y = 100000 + \frac{50000 x}{7} + \frac{10000 x^{2}}{49} + 1000 \frac{x^{3}}{343} + 5 \cdot 10 {(\frac{x}{7})}^{4} + {(\frac{x}{7})}^{5}$

解题步骤 3.5.2.1.2.13

组合 $1000$ 和 $\frac{x^{3}}{343}$ 。

$y = 100000 + \frac{50000 x}{7} + \frac{10000 x^{2}}{49} + \frac{1000 x^{3}}{343} + 5 \cdot 10 {(\frac{x}{7})}^{4} + {(\frac{x}{7})}^{5}$

解题步骤 3.5.2.1.2.14

将 $5$ 乘以 $10$ 。

$y = 100000 + \frac{50000 x}{7} + \frac{10000 x^{2}}{49} + \frac{1000 x^{3}}{343} + 50 {(\frac{x}{7})}^{4} + {(\frac{x}{7})}^{5}$

解题步骤 3.5.2.1.2.15

对 $\frac{x}{7}$ 运用乘积法则。

$y = 100000 + \frac{50000 x}{7} + \frac{10000 x^{2}}{49} + \frac{1000 x^{3}}{343} + 50 \frac{x^{4}}{7^{4}} + {(\frac{x}{7})}^{5}$

解题步骤 3.5.2.1.2.16

对 $7$ 进行 $4$ 次方运算。

$y = 100000 + \frac{50000 x}{7} + \frac{10000 x^{2}}{49} + \frac{1000 x^{3}}{343} + 50 \frac{x^{4}}{2401} + {(\frac{x}{7})}^{5}$

解题步骤 3.5.2.1.2.17

组合 $50$ 和 $\frac{x^{4}}{2401}$ 。

$y = 100000 + \frac{50000 x}{7} + \frac{10000 x^{2}}{49} + \frac{1000 x^{3}}{343} + \frac{50 x^{4}}{2401} + {(\frac{x}{7})}^{5}$

解题步骤 3.5.2.1.2.18

对 $\frac{x}{7}$ 运用乘积法则。

$y = 100000 + \frac{50000 x}{7} + \frac{10000 x^{2}}{49} + \frac{1000 x^{3}}{343} + \frac{50 x^{4}}{2401} + \frac{x^{5}}{7^{5}}$

解题步骤 3.5.2.1.2.19

对 $7$ 进行 $5$ 次方运算。

$y = 100000 + \frac{50000 x}{7} + \frac{10000 x^{2}}{49} + \frac{1000 x^{3}}{343} + \frac{50 x^{4}}{2401} + \frac{x^{5}}{16807}$

解题步骤 3.6

化简 $100000 + \frac{50000 x}{7} + \frac{10000 x^{2}}{49} + \frac{1000 x^{3}}{343} + \frac{50 x^{4}}{2401} + \frac{x^{5}}{16807}$ 。

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解题步骤 3.6.1

移动 $100000$ 。

$y = \frac{50000 x}{7} + \frac{10000 x^{2}}{49} + \frac{1000 x^{3}}{343} + \frac{50 x^{4}}{2401} + \frac{x^{5}}{16807} + 100000$

解题步骤 3.6.2

移动 $\frac{50000 x}{7}$ 。

$y = \frac{10000 x^{2}}{49} + \frac{1000 x^{3}}{343} + \frac{50 x^{4}}{2401} + \frac{x^{5}}{16807} + \frac{50000 x}{7} + 100000$

解题步骤 3.6.3

移动 $\frac{10000 x^{2}}{49}$ 。

$y = \frac{1000 x^{3}}{343} + \frac{50 x^{4}}{2401} + \frac{x^{5}}{16807} + \frac{10000 x^{2}}{49} + \frac{50000 x}{7} + 100000$

解题步骤 3.6.4

移动 $\frac{1000 x^{3}}{343}$ 。

$y = \frac{50 x^{4}}{2401} + \frac{x^{5}}{16807} + \frac{1000 x^{3}}{343} + \frac{10000 x^{2}}{49} + \frac{50000 x}{7} + 100000$

解题步骤 3.6.5

将 $\frac{50 x^{4}}{2401}$ 和 $\frac{x^{5}}{16807}$ 重新排序。

$y = \frac{x^{5}}{16807} + \frac{50 x^{4}}{2401} + \frac{1000 x^{3}}{343} + \frac{10000 x^{2}}{49} + \frac{50000 x}{7} + 100000$

解题步骤 4

使用 $f^{- 1} (x)$ 替换 $y$ ，以得到最终答案。

$f^{- 1} (x) = \frac{x^{5}}{16807} + \frac{50 x^{4}}{2401} + \frac{1000 x^{3}}{343} + \frac{10000 x^{2}}{49} + \frac{50000 x}{7} + 100000$

解题步骤 5

验证 $f^{- 1} (x) = \frac{x^{5}}{16807} + \frac{50 x^{4}}{2401} + \frac{1000 x^{3}}{343} + \frac{10000 x^{2}}{49} + \frac{50000 x}{7} + 100000$ 是否为 $f (x) = 7 (x^{\frac{1}{5}} - 10)$ 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 $f^{- 1} (7 (x^{\frac{1}{5}} - 10))$ 。

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = \frac{{(7 (x^{\frac{1}{5}} - 10))}^{5}}{16807} + \frac{50 {(7 (x^{\frac{1}{5}} - 10))}^{4}}{2401} + \frac{1000 {(7 (x^{\frac{1}{5}} - 10))}^{3}}{343} + \frac{10000 {(7 (x^{\frac{1}{5}} - 10))}^{2}}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

解题步骤 5.2.3

化简每一项。

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解题步骤 5.2.3.1

化简分子。

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解题步骤 5.2.3.1.1

对 $7 (x^{\frac{1}{5}} - 10)$ 运用乘积法则。

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = \frac{7^{5} {(x^{\frac{1}{5}} - 10)}^{5}}{16807} + \frac{50 {(7 (x^{\frac{1}{5}} - 10))}^{4}}{2401} + \frac{1000 {(7 (x^{\frac{1}{5}} - 10))}^{3}}{343} + \frac{10000 {(7 (x^{\frac{1}{5}} - 10))}^{2}}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

解题步骤 5.2.3.1.2

对 $7$ 进行 $5$ 次方运算。

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = \frac{16807 {(x^{\frac{1}{5}} - 10)}^{5}}{16807} + \frac{50 {(7 (x^{\frac{1}{5}} - 10))}^{4}}{2401} + \frac{1000 {(7 (x^{\frac{1}{5}} - 10))}^{3}}{343} + \frac{10000 {(7 (x^{\frac{1}{5}} - 10))}^{2}}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

解题步骤 5.2.3.2

约去公因数。

解题步骤 5.2.3.3

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

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = {(x^{\frac{1}{5}} - 10)}^{5} + \frac{50 {(7 (x^{\frac{1}{5}} - 10))}^{4}}{2401} + \frac{1000 {(7 (x^{\frac{1}{5}} - 10))}^{3}}{343} + \frac{10000 {(7 (x^{\frac{1}{5}} - 10))}^{2}}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

解题步骤 5.2.3.4

使用二项式定理。

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = {(x^{\frac{1}{5}})}^{5} + 5 {(x^{\frac{1}{5}})}^{4} \cdot - 10 + 10 {(x^{\frac{1}{5}})}^{3} {(- 10)}^{2} + 10 {(x^{\frac{1}{5}})}^{2} {(- 10)}^{3} + 5 x^{\frac{1}{5}} {(- 10)}^{4} + {(- 10)}^{5} + \frac{50 {(7 (x^{\frac{1}{5}} - 10))}^{4}}{2401} + \frac{1000 {(7 (x^{\frac{1}{5}} - 10))}^{3}}{343} + \frac{10000 {(7 (x^{\frac{1}{5}} - 10))}^{2}}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

解题步骤 5.2.3.5

化简每一项。

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解题步骤 5.2.3.5.1

将 ${(x^{\frac{1}{5}})}^{5}$ 中的指数相乘。

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解题步骤 5.2.3.5.1.1

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

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x^{\frac{1}{5} \cdot 5} + 5 {(x^{\frac{1}{5}})}^{4} \cdot - 10 + 10 {(x^{\frac{1}{5}})}^{3} {(- 10)}^{2} + 10 {(x^{\frac{1}{5}})}^{2} {(- 10)}^{3} + 5 x^{\frac{1}{5}} {(- 10)}^{4} + {(- 10)}^{5} + \frac{50 {(7 (x^{\frac{1}{5}} - 10))}^{4}}{2401} + \frac{1000 {(7 (x^{\frac{1}{5}} - 10))}^{3}}{343} + \frac{10000 {(7 (x^{\frac{1}{5}} - 10))}^{2}}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

解题步骤 5.2.3.5.1.2

约去 $5$ 的公因数。

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解题步骤 5.2.3.5.1.2.1

约去公因数。

解题步骤 5.2.3.5.1.2.2

重写表达式。

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x + 5 {(x^{\frac{1}{5}})}^{4} \cdot - 10 + 10 {(x^{\frac{1}{5}})}^{3} {(- 10)}^{2} + 10 {(x^{\frac{1}{5}})}^{2} {(- 10)}^{3} + 5 x^{\frac{1}{5}} {(- 10)}^{4} + {(- 10)}^{5} + \frac{50 {(7 (x^{\frac{1}{5}} - 10))}^{4}}{2401} + \frac{1000 {(7 (x^{\frac{1}{5}} - 10))}^{3}}{343} + \frac{10000 {(7 (x^{\frac{1}{5}} - 10))}^{2}}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

解题步骤 5.2.3.5.2

化简。

解题步骤 5.2.3.5.3

将 ${(x^{\frac{1}{5}})}^{4}$ 中的指数相乘。

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解题步骤 5.2.3.5.3.1

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

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x + 5 x^{\frac{1}{5} \cdot 4} \cdot - 10 + 10 {(x^{\frac{1}{5}})}^{3} {(- 10)}^{2} + 10 {(x^{\frac{1}{5}})}^{2} {(- 10)}^{3} + 5 x^{\frac{1}{5}} {(- 10)}^{4} + {(- 10)}^{5} + \frac{50 {(7 (x^{\frac{1}{5}} - 10))}^{4}}{2401} + \frac{1000 {(7 (x^{\frac{1}{5}} - 10))}^{3}}{343} + \frac{10000 {(7 (x^{\frac{1}{5}} - 10))}^{2}}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

解题步骤 5.2.3.5.3.2

组合 $\frac{1}{5}$ 和 $4$ 。

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x + 5 x^{\frac{4}{5}} \cdot - 10 + 10 {(x^{\frac{1}{5}})}^{3} {(- 10)}^{2} + 10 {(x^{\frac{1}{5}})}^{2} {(- 10)}^{3} + 5 x^{\frac{1}{5}} {(- 10)}^{4} + {(- 10)}^{5} + \frac{50 {(7 (x^{\frac{1}{5}} - 10))}^{4}}{2401} + \frac{1000 {(7 (x^{\frac{1}{5}} - 10))}^{3}}{343} + \frac{10000 {(7 (x^{\frac{1}{5}} - 10))}^{2}}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

解题步骤 5.2.3.5.4

将 $- 10$ 乘以 $5$ 。

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 10 {(x^{\frac{1}{5}})}^{3} {(- 10)}^{2} + 10 {(x^{\frac{1}{5}})}^{2} {(- 10)}^{3} + 5 x^{\frac{1}{5}} {(- 10)}^{4} + {(- 10)}^{5} + \frac{50 {(7 (x^{\frac{1}{5}} - 10))}^{4}}{2401} + \frac{1000 {(7 (x^{\frac{1}{5}} - 10))}^{3}}{343} + \frac{10000 {(7 (x^{\frac{1}{5}} - 10))}^{2}}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

解题步骤 5.2.3.5.5

将 ${(x^{\frac{1}{5}})}^{3}$ 中的指数相乘。

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解题步骤 5.2.3.5.5.1

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

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 10 x^{\frac{1}{5} \cdot 3} {(- 10)}^{2} + 10 {(x^{\frac{1}{5}})}^{2} {(- 10)}^{3} + 5 x^{\frac{1}{5}} {(- 10)}^{4} + {(- 10)}^{5} + \frac{50 {(7 (x^{\frac{1}{5}} - 10))}^{4}}{2401} + \frac{1000 {(7 (x^{\frac{1}{5}} - 10))}^{3}}{343} + \frac{10000 {(7 (x^{\frac{1}{5}} - 10))}^{2}}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

解题步骤 5.2.3.5.5.2

组合 $\frac{1}{5}$ 和 $3$ 。

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 10 x^{\frac{3}{5}} {(- 10)}^{2} + 10 {(x^{\frac{1}{5}})}^{2} {(- 10)}^{3} + 5 x^{\frac{1}{5}} {(- 10)}^{4} + {(- 10)}^{5} + \frac{50 {(7 (x^{\frac{1}{5}} - 10))}^{4}}{2401} + \frac{1000 {(7 (x^{\frac{1}{5}} - 10))}^{3}}{343} + \frac{10000 {(7 (x^{\frac{1}{5}} - 10))}^{2}}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

解题步骤 5.2.3.5.6

对 $- 10$ 进行 $2$ 次方运算。

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 10 x^{\frac{3}{5}} \cdot 100 + 10 {(x^{\frac{1}{5}})}^{2} {(- 10)}^{3} + 5 x^{\frac{1}{5}} {(- 10)}^{4} + {(- 10)}^{5} + \frac{50 {(7 (x^{\frac{1}{5}} - 10))}^{4}}{2401} + \frac{1000 {(7 (x^{\frac{1}{5}} - 10))}^{3}}{343} + \frac{10000 {(7 (x^{\frac{1}{5}} - 10))}^{2}}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

解题步骤 5.2.3.5.7

将 $100$ 乘以 $10$ 。

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} + 10 {(x^{\frac{1}{5}})}^{2} {(- 10)}^{3} + 5 x^{\frac{1}{5}} {(- 10)}^{4} + {(- 10)}^{5} + \frac{50 {(7 (x^{\frac{1}{5}} - 10))}^{4}}{2401} + \frac{1000 {(7 (x^{\frac{1}{5}} - 10))}^{3}}{343} + \frac{10000 {(7 (x^{\frac{1}{5}} - 10))}^{2}}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

解题步骤 5.2.3.5.8

将 ${(x^{\frac{1}{5}})}^{2}$ 中的指数相乘。

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解题步骤 5.2.3.5.8.1

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

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} + 10 x^{\frac{1}{5} \cdot 2} {(- 10)}^{3} + 5 x^{\frac{1}{5}} {(- 10)}^{4} + {(- 10)}^{5} + \frac{50 {(7 (x^{\frac{1}{5}} - 10))}^{4}}{2401} + \frac{1000 {(7 (x^{\frac{1}{5}} - 10))}^{3}}{343} + \frac{10000 {(7 (x^{\frac{1}{5}} - 10))}^{2}}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

解题步骤 5.2.3.5.8.2

组合 $\frac{1}{5}$ 和 $2$ 。

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} + 10 x^{\frac{2}{5}} {(- 10)}^{3} + 5 x^{\frac{1}{5}} {(- 10)}^{4} + {(- 10)}^{5} + \frac{50 {(7 (x^{\frac{1}{5}} - 10))}^{4}}{2401} + \frac{1000 {(7 (x^{\frac{1}{5}} - 10))}^{3}}{343} + \frac{10000 {(7 (x^{\frac{1}{5}} - 10))}^{2}}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

解题步骤 5.2.3.5.9

对 $- 10$ 进行 $3$ 次方运算。

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} + 10 x^{\frac{2}{5}} \cdot - 1000 + 5 x^{\frac{1}{5}} {(- 10)}^{4} + {(- 10)}^{5} + \frac{50 {(7 (x^{\frac{1}{5}} - 10))}^{4}}{2401} + \frac{1000 {(7 (x^{\frac{1}{5}} - 10))}^{3}}{343} + \frac{10000 {(7 (x^{\frac{1}{5}} - 10))}^{2}}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

解题步骤 5.2.3.5.10

将 $- 1000$ 乘以 $10$ 。

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 5 x^{\frac{1}{5}} {(- 10)}^{4} + {(- 10)}^{5} + \frac{50 {(7 (x^{\frac{1}{5}} - 10))}^{4}}{2401} + \frac{1000 {(7 (x^{\frac{1}{5}} - 10))}^{3}}{343} + \frac{10000 {(7 (x^{\frac{1}{5}} - 10))}^{2}}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

解题步骤 5.2.3.5.11

对 $- 10$ 进行 $4$ 次方运算。

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 5 x^{\frac{1}{5}} \cdot 10000 + {(- 10)}^{5} + \frac{50 {(7 (x^{\frac{1}{5}} - 10))}^{4}}{2401} + \frac{1000 {(7 (x^{\frac{1}{5}} - 10))}^{3}}{343} + \frac{10000 {(7 (x^{\frac{1}{5}} - 10))}^{2}}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

解题步骤 5.2.3.5.12

将 $10000$ 乘以 $5$ 。

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} + {(- 10)}^{5} + \frac{50 {(7 (x^{\frac{1}{5}} - 10))}^{4}}{2401} + \frac{1000 {(7 (x^{\frac{1}{5}} - 10))}^{3}}{343} + \frac{10000 {(7 (x^{\frac{1}{5}} - 10))}^{2}}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

解题步骤 5.2.3.5.13

对 $- 10$ 进行 $5$ 次方运算。

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + \frac{50 {(7 (x^{\frac{1}{5}} - 10))}^{4}}{2401} + \frac{1000 {(7 (x^{\frac{1}{5}} - 10))}^{3}}{343} + \frac{10000 {(7 (x^{\frac{1}{5}} - 10))}^{2}}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

解题步骤 5.2.3.6

化简分子。

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解题步骤 5.2.3.6.1

对 $7 (x^{\frac{1}{5}} - 10)$ 运用乘积法则。

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + \frac{50 \cdot (7^{4} {(x^{\frac{1}{5}} - 10)}^{4})}{2401} + \frac{1000 {(7 (x^{\frac{1}{5}} - 10))}^{3}}{343} + \frac{10000 {(7 (x^{\frac{1}{5}} - 10))}^{2}}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

解题步骤 5.2.3.6.2

对 $7$ 进行 $4$ 次方运算。

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + \frac{50 \cdot (2401 {(x^{\frac{1}{5}} - 10)}^{4})}{2401} + \frac{1000 {(7 (x^{\frac{1}{5}} - 10))}^{3}}{343} + \frac{10000 {(7 (x^{\frac{1}{5}} - 10))}^{2}}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

解题步骤 5.2.3.6.3

将 $50$ 乘以 $2401$ 。

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + \frac{120050 {(x^{\frac{1}{5}} - 10)}^{4}}{2401} + \frac{1000 {(7 (x^{\frac{1}{5}} - 10))}^{3}}{343} + \frac{10000 {(7 (x^{\frac{1}{5}} - 10))}^{2}}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

解题步骤 5.2.3.7

从 $120050 {(x^{\frac{1}{5}} - 10)}^{4}$ 中分解出因数 $2401$ 。

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + \frac{2401 (50 {(x^{\frac{1}{5}} - 10)}^{4})}{2401} + \frac{1000 {(7 (x^{\frac{1}{5}} - 10))}^{3}}{343} + \frac{10000 {(7 (x^{\frac{1}{5}} - 10))}^{2}}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

解题步骤 5.2.3.8

约去公因数。

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解题步骤 5.2.3.8.1

从 $2401$ 中分解出因数 $2401$ 。

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + \frac{2401 (50 {(x^{\frac{1}{5}} - 10)}^{4})}{2401 (1)} + \frac{1000 {(7 (x^{\frac{1}{5}} - 10))}^{3}}{343} + \frac{10000 {(7 (x^{\frac{1}{5}} - 10))}^{2}}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

解题步骤 5.2.3.8.2

约去公因数。

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + \frac{2401 (50 {(x^{\frac{1}{5}} - 10)}^{4})}{2401 \cdot 1} + \frac{1000 {(7 (x^{\frac{1}{5}} - 10))}^{3}}{343} + \frac{10000 {(7 (x^{\frac{1}{5}} - 10))}^{2}}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

解题步骤 5.2.3.8.3

重写表达式。

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + \frac{50 {(x^{\frac{1}{5}} - 10)}^{4}}{1} + \frac{1000 {(7 (x^{\frac{1}{5}} - 10))}^{3}}{343} + \frac{10000 {(7 (x^{\frac{1}{5}} - 10))}^{2}}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

解题步骤 5.2.3.8.4

用 $50 {(x^{\frac{1}{5}} - 10)}^{4}$ 除以 $1$ 。

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + 50 {(x^{\frac{1}{5}} - 10)}^{4} + \frac{1000 {(7 (x^{\frac{1}{5}} - 10))}^{3}}{343} + \frac{10000 {(7 (x^{\frac{1}{5}} - 10))}^{2}}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

解题步骤 5.2.3.9

使用二项式定理。

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + 50 ({(x^{\frac{1}{5}})}^{4} + 4 {(x^{\frac{1}{5}})}^{3} \cdot - 10 + 6 {(x^{\frac{1}{5}})}^{2} {(- 10)}^{2} + 4 x^{\frac{1}{5}} {(- 10)}^{3} + {(- 10)}^{4}) + \frac{1000 {(7 (x^{\frac{1}{5}} - 10))}^{3}}{343} + \frac{10000 {(7 (x^{\frac{1}{5}} - 10))}^{2}}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

解题步骤 5.2.3.10

化简每一项。

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解题步骤 5.2.3.10.1

将 ${(x^{\frac{1}{5}})}^{4}$ 中的指数相乘。

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解题步骤 5.2.3.10.1.1

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

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + 50 (x^{\frac{1}{5} \cdot 4} + 4 {(x^{\frac{1}{5}})}^{3} \cdot - 10 + 6 {(x^{\frac{1}{5}})}^{2} {(- 10)}^{2} + 4 x^{\frac{1}{5}} {(- 10)}^{3} + {(- 10)}^{4}) + \frac{1000 {(7 (x^{\frac{1}{5}} - 10))}^{3}}{343} + \frac{10000 {(7 (x^{\frac{1}{5}} - 10))}^{2}}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

解题步骤 5.2.3.10.1.2

组合 $\frac{1}{5}$ 和 $4$ 。

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + 50 (x^{\frac{4}{5}} + 4 {(x^{\frac{1}{5}})}^{3} \cdot - 10 + 6 {(x^{\frac{1}{5}})}^{2} {(- 10)}^{2} + 4 x^{\frac{1}{5}} {(- 10)}^{3} + {(- 10)}^{4}) + \frac{1000 {(7 (x^{\frac{1}{5}} - 10))}^{3}}{343} + \frac{10000 {(7 (x^{\frac{1}{5}} - 10))}^{2}}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

解题步骤 5.2.3.10.2

将 ${(x^{\frac{1}{5}})}^{3}$ 中的指数相乘。

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解题步骤 5.2.3.10.2.1

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

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + 50 (x^{\frac{4}{5}} + 4 x^{\frac{1}{5} \cdot 3} \cdot - 10 + 6 {(x^{\frac{1}{5}})}^{2} {(- 10)}^{2} + 4 x^{\frac{1}{5}} {(- 10)}^{3} + {(- 10)}^{4}) + \frac{1000 {(7 (x^{\frac{1}{5}} - 10))}^{3}}{343} + \frac{10000 {(7 (x^{\frac{1}{5}} - 10))}^{2}}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

解题步骤 5.2.3.10.2.2

组合 $\frac{1}{5}$ 和 $3$ 。

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + 50 (x^{\frac{4}{5}} + 4 x^{\frac{3}{5}} \cdot - 10 + 6 {(x^{\frac{1}{5}})}^{2} {(- 10)}^{2} + 4 x^{\frac{1}{5}} {(- 10)}^{3} + {(- 10)}^{4}) + \frac{1000 {(7 (x^{\frac{1}{5}} - 10))}^{3}}{343} + \frac{10000 {(7 (x^{\frac{1}{5}} - 10))}^{2}}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

解题步骤 5.2.3.10.3

将 $- 10$ 乘以 $4$ 。

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + 50 (x^{\frac{4}{5}} - 40 x^{\frac{3}{5}} + 6 {(x^{\frac{1}{5}})}^{2} {(- 10)}^{2} + 4 x^{\frac{1}{5}} {(- 10)}^{3} + {(- 10)}^{4}) + \frac{1000 {(7 (x^{\frac{1}{5}} - 10))}^{3}}{343} + \frac{10000 {(7 (x^{\frac{1}{5}} - 10))}^{2}}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

解题步骤 5.2.3.10.4

将 ${(x^{\frac{1}{5}})}^{2}$ 中的指数相乘。

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解题步骤 5.2.3.10.4.1

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

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + 50 (x^{\frac{4}{5}} - 40 x^{\frac{3}{5}} + 6 x^{\frac{1}{5} \cdot 2} {(- 10)}^{2} + 4 x^{\frac{1}{5}} {(- 10)}^{3} + {(- 10)}^{4}) + \frac{1000 {(7 (x^{\frac{1}{5}} - 10))}^{3}}{343} + \frac{10000 {(7 (x^{\frac{1}{5}} - 10))}^{2}}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

解题步骤 5.2.3.10.4.2

组合 $\frac{1}{5}$ 和 $2$ 。

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + 50 (x^{\frac{4}{5}} - 40 x^{\frac{3}{5}} + 6 x^{\frac{2}{5}} {(- 10)}^{2} + 4 x^{\frac{1}{5}} {(- 10)}^{3} + {(- 10)}^{4}) + \frac{1000 {(7 (x^{\frac{1}{5}} - 10))}^{3}}{343} + \frac{10000 {(7 (x^{\frac{1}{5}} - 10))}^{2}}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

解题步骤 5.2.3.10.5

对 $- 10$ 进行 $2$ 次方运算。

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + 50 (x^{\frac{4}{5}} - 40 x^{\frac{3}{5}} + 6 x^{\frac{2}{5}} \cdot 100 + 4 x^{\frac{1}{5}} {(- 10)}^{3} + {(- 10)}^{4}) + \frac{1000 {(7 (x^{\frac{1}{5}} - 10))}^{3}}{343} + \frac{10000 {(7 (x^{\frac{1}{5}} - 10))}^{2}}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

解题步骤 5.2.3.10.6

将 $100$ 乘以 $6$ 。

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + 50 (x^{\frac{4}{5}} - 40 x^{\frac{3}{5}} + 600 x^{\frac{2}{5}} + 4 x^{\frac{1}{5}} {(- 10)}^{3} + {(- 10)}^{4}) + \frac{1000 {(7 (x^{\frac{1}{5}} - 10))}^{3}}{343} + \frac{10000 {(7 (x^{\frac{1}{5}} - 10))}^{2}}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

解题步骤 5.2.3.10.7

对 $- 10$ 进行 $3$ 次方运算。

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + 50 (x^{\frac{4}{5}} - 40 x^{\frac{3}{5}} + 600 x^{\frac{2}{5}} + 4 x^{\frac{1}{5}} \cdot - 1000 + {(- 10)}^{4}) + \frac{1000 {(7 (x^{\frac{1}{5}} - 10))}^{3}}{343} + \frac{10000 {(7 (x^{\frac{1}{5}} - 10))}^{2}}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

解题步骤 5.2.3.10.8

将 $- 1000$ 乘以 $4$ 。

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + 50 (x^{\frac{4}{5}} - 40 x^{\frac{3}{5}} + 600 x^{\frac{2}{5}} - 4000 x^{\frac{1}{5}} + {(- 10)}^{4}) + \frac{1000 {(7 (x^{\frac{1}{5}} - 10))}^{3}}{343} + \frac{10000 {(7 (x^{\frac{1}{5}} - 10))}^{2}}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

解题步骤 5.2.3.10.9

对 $- 10$ 进行 $4$ 次方运算。

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + 50 (x^{\frac{4}{5}} - 40 x^{\frac{3}{5}} + 600 x^{\frac{2}{5}} - 4000 x^{\frac{1}{5}} + 10000) + \frac{1000 {(7 (x^{\frac{1}{5}} - 10))}^{3}}{343} + \frac{10000 {(7 (x^{\frac{1}{5}} - 10))}^{2}}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

解题步骤 5.2.3.11

运用分配律。

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + 50 x^{\frac{4}{5}} + 50 (- 40 x^{\frac{3}{5}}) + 50 (600 x^{\frac{2}{5}}) + 50 (- 4000 x^{\frac{1}{5}}) + 50 \cdot 10000 + \frac{1000 {(7 (x^{\frac{1}{5}} - 10))}^{3}}{343} + \frac{10000 {(7 (x^{\frac{1}{5}} - 10))}^{2}}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

解题步骤 5.2.3.12

化简。

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解题步骤 5.2.3.12.1

将 $- 40$ 乘以 $50$ 。

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + 50 x^{\frac{4}{5}} - 2000 x^{\frac{3}{5}} + 50 (600 x^{\frac{2}{5}}) + 50 (- 4000 x^{\frac{1}{5}}) + 50 \cdot 10000 + \frac{1000 {(7 (x^{\frac{1}{5}} - 10))}^{3}}{343} + \frac{10000 {(7 (x^{\frac{1}{5}} - 10))}^{2}}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

解题步骤 5.2.3.12.2

将 $600$ 乘以 $50$ 。

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + 50 x^{\frac{4}{5}} - 2000 x^{\frac{3}{5}} + 30000 x^{\frac{2}{5}} + 50 (- 4000 x^{\frac{1}{5}}) + 50 \cdot 10000 + \frac{1000 {(7 (x^{\frac{1}{5}} - 10))}^{3}}{343} + \frac{10000 {(7 (x^{\frac{1}{5}} - 10))}^{2}}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

解题步骤 5.2.3.12.3

将 $- 4000$ 乘以 $50$ 。

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + 50 x^{\frac{4}{5}} - 2000 x^{\frac{3}{5}} + 30000 x^{\frac{2}{5}} - 200000 x^{\frac{1}{5}} + 50 \cdot 10000 + \frac{1000 {(7 (x^{\frac{1}{5}} - 10))}^{3}}{343} + \frac{10000 {(7 (x^{\frac{1}{5}} - 10))}^{2}}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

解题步骤 5.2.3.12.4

将 $50$ 乘以 $10000$ 。

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + 50 x^{\frac{4}{5}} - 2000 x^{\frac{3}{5}} + 30000 x^{\frac{2}{5}} - 200000 x^{\frac{1}{5}} + 500000 + \frac{1000 {(7 (x^{\frac{1}{5}} - 10))}^{3}}{343} + \frac{10000 {(7 (x^{\frac{1}{5}} - 10))}^{2}}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

解题步骤 5.2.3.13

化简分子。

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解题步骤 5.2.3.13.1

对 $7 (x^{\frac{1}{5}} - 10)$ 运用乘积法则。

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + 50 x^{\frac{4}{5}} - 2000 x^{\frac{3}{5}} + 30000 x^{\frac{2}{5}} - 200000 x^{\frac{1}{5}} + 500000 + \frac{1000 \cdot (7^{3} {(x^{\frac{1}{5}} - 10)}^{3})}{343} + \frac{10000 {(7 (x^{\frac{1}{5}} - 10))}^{2}}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

解题步骤 5.2.3.13.2

对 $7$ 进行 $3$ 次方运算。

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + 50 x^{\frac{4}{5}} - 2000 x^{\frac{3}{5}} + 30000 x^{\frac{2}{5}} - 200000 x^{\frac{1}{5}} + 500000 + \frac{1000 \cdot (343 {(x^{\frac{1}{5}} - 10)}^{3})}{343} + \frac{10000 {(7 (x^{\frac{1}{5}} - 10))}^{2}}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

解题步骤 5.2.3.13.3

将 $1000$ 乘以 $343$ 。

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + 50 x^{\frac{4}{5}} - 2000 x^{\frac{3}{5}} + 30000 x^{\frac{2}{5}} - 200000 x^{\frac{1}{5}} + 500000 + \frac{343000 {(x^{\frac{1}{5}} - 10)}^{3}}{343} + \frac{10000 {(7 (x^{\frac{1}{5}} - 10))}^{2}}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

解题步骤 5.2.3.14

从 $343000 {(x^{\frac{1}{5}} - 10)}^{3}$ 中分解出因数 $343$ 。

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + 50 x^{\frac{4}{5}} - 2000 x^{\frac{3}{5}} + 30000 x^{\frac{2}{5}} - 200000 x^{\frac{1}{5}} + 500000 + \frac{343 (1000 {(x^{\frac{1}{5}} - 10)}^{3})}{343} + \frac{10000 {(7 (x^{\frac{1}{5}} - 10))}^{2}}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

解题步骤 5.2.3.15

约去公因数。

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解题步骤 5.2.3.15.1

从 $343$ 中分解出因数 $343$ 。

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + 50 x^{\frac{4}{5}} - 2000 x^{\frac{3}{5}} + 30000 x^{\frac{2}{5}} - 200000 x^{\frac{1}{5}} + 500000 + \frac{343 (1000 {(x^{\frac{1}{5}} - 10)}^{3})}{343 (1)} + \frac{10000 {(7 (x^{\frac{1}{5}} - 10))}^{2}}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

解题步骤 5.2.3.15.2

约去公因数。

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + 50 x^{\frac{4}{5}} - 2000 x^{\frac{3}{5}} + 30000 x^{\frac{2}{5}} - 200000 x^{\frac{1}{5}} + 500000 + \frac{343 (1000 {(x^{\frac{1}{5}} - 10)}^{3})}{343 \cdot 1} + \frac{10000 {(7 (x^{\frac{1}{5}} - 10))}^{2}}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

解题步骤 5.2.3.15.3

重写表达式。

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + 50 x^{\frac{4}{5}} - 2000 x^{\frac{3}{5}} + 30000 x^{\frac{2}{5}} - 200000 x^{\frac{1}{5}} + 500000 + \frac{1000 {(x^{\frac{1}{5}} - 10)}^{3}}{1} + \frac{10000 {(7 (x^{\frac{1}{5}} - 10))}^{2}}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

解题步骤 5.2.3.15.4

用 $1000 {(x^{\frac{1}{5}} - 10)}^{3}$ 除以 $1$ 。

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + 50 x^{\frac{4}{5}} - 2000 x^{\frac{3}{5}} + 30000 x^{\frac{2}{5}} - 200000 x^{\frac{1}{5}} + 500000 + 1000 {(x^{\frac{1}{5}} - 10)}^{3} + \frac{10000 {(7 (x^{\frac{1}{5}} - 10))}^{2}}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

解题步骤 5.2.3.16

使用二项式定理。

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + 50 x^{\frac{4}{5}} - 2000 x^{\frac{3}{5}} + 30000 x^{\frac{2}{5}} - 200000 x^{\frac{1}{5}} + 500000 + 1000 ({(x^{\frac{1}{5}})}^{3} + 3 {(x^{\frac{1}{5}})}^{2} \cdot - 10 + 3 x^{\frac{1}{5}} {(- 10)}^{2} + {(- 10)}^{3}) + \frac{10000 {(7 (x^{\frac{1}{5}} - 10))}^{2}}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

解题步骤 5.2.3.17

化简每一项。

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解题步骤 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 5.2.3.17.1.1

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

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + 50 x^{\frac{4}{5}} - 2000 x^{\frac{3}{5}} + 30000 x^{\frac{2}{5}} - 200000 x^{\frac{1}{5}} + 500000 + 1000 (x^{\frac{1}{5} \cdot 3} + 3 {(x^{\frac{1}{5}})}^{2} \cdot - 10 + 3 x^{\frac{1}{5}} {(- 10)}^{2} + {(- 10)}^{3}) + \frac{10000 {(7 (x^{\frac{1}{5}} - 10))}^{2}}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

解题步骤 5.2.3.17.1.2

组合 $\frac{1}{5}$ 和 $3$ 。

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + 50 x^{\frac{4}{5}} - 2000 x^{\frac{3}{5}} + 30000 x^{\frac{2}{5}} - 200000 x^{\frac{1}{5}} + 500000 + 1000 (x^{\frac{3}{5}} + 3 {(x^{\frac{1}{5}})}^{2} \cdot - 10 + 3 x^{\frac{1}{5}} {(- 10)}^{2} + {(- 10)}^{3}) + \frac{10000 {(7 (x^{\frac{1}{5}} - 10))}^{2}}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

解题步骤 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 5.2.3.17.2.1

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

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + 50 x^{\frac{4}{5}} - 2000 x^{\frac{3}{5}} + 30000 x^{\frac{2}{5}} - 200000 x^{\frac{1}{5}} + 500000 + 1000 (x^{\frac{3}{5}} + 3 x^{\frac{1}{5} \cdot 2} \cdot - 10 + 3 x^{\frac{1}{5}} {(- 10)}^{2} + {(- 10)}^{3}) + \frac{10000 {(7 (x^{\frac{1}{5}} - 10))}^{2}}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

解题步骤 5.2.3.17.2.2

组合 $\frac{1}{5}$ 和 $2$ 。

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + 50 x^{\frac{4}{5}} - 2000 x^{\frac{3}{5}} + 30000 x^{\frac{2}{5}} - 200000 x^{\frac{1}{5}} + 500000 + 1000 (x^{\frac{3}{5}} + 3 x^{\frac{2}{5}} \cdot - 10 + 3 x^{\frac{1}{5}} {(- 10)}^{2} + {(- 10)}^{3}) + \frac{10000 {(7 (x^{\frac{1}{5}} - 10))}^{2}}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

解题步骤 5.2.3.17.3

将 $- 10$ 乘以 $3$ 。

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + 50 x^{\frac{4}{5}} - 2000 x^{\frac{3}{5}} + 30000 x^{\frac{2}{5}} - 200000 x^{\frac{1}{5}} + 500000 + 1000 (x^{\frac{3}{5}} - 30 x^{\frac{2}{5}} + 3 x^{\frac{1}{5}} {(- 10)}^{2} + {(- 10)}^{3}) + \frac{10000 {(7 (x^{\frac{1}{5}} - 10))}^{2}}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

解题步骤 5.2.3.17.4

对 $- 10$ 进行 $2$ 次方运算。

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + 50 x^{\frac{4}{5}} - 2000 x^{\frac{3}{5}} + 30000 x^{\frac{2}{5}} - 200000 x^{\frac{1}{5}} + 500000 + 1000 (x^{\frac{3}{5}} - 30 x^{\frac{2}{5}} + 3 x^{\frac{1}{5}} \cdot 100 + {(- 10)}^{3}) + \frac{10000 {(7 (x^{\frac{1}{5}} - 10))}^{2}}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

解题步骤 5.2.3.17.5

将 $100$ 乘以 $3$ 。

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + 50 x^{\frac{4}{5}} - 2000 x^{\frac{3}{5}} + 30000 x^{\frac{2}{5}} - 200000 x^{\frac{1}{5}} + 500000 + 1000 (x^{\frac{3}{5}} - 30 x^{\frac{2}{5}} + 300 x^{\frac{1}{5}} + {(- 10)}^{3}) + \frac{10000 {(7 (x^{\frac{1}{5}} - 10))}^{2}}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

解题步骤 5.2.3.17.6

对 $- 10$ 进行 $3$ 次方运算。

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + 50 x^{\frac{4}{5}} - 2000 x^{\frac{3}{5}} + 30000 x^{\frac{2}{5}} - 200000 x^{\frac{1}{5}} + 500000 + 1000 (x^{\frac{3}{5}} - 30 x^{\frac{2}{5}} + 300 x^{\frac{1}{5}} - 1000) + \frac{10000 {(7 (x^{\frac{1}{5}} - 10))}^{2}}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

解题步骤 5.2.3.18

运用分配律。

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + 50 x^{\frac{4}{5}} - 2000 x^{\frac{3}{5}} + 30000 x^{\frac{2}{5}} - 200000 x^{\frac{1}{5}} + 500000 + 1000 x^{\frac{3}{5}} + 1000 (- 30 x^{\frac{2}{5}}) + 1000 (300 x^{\frac{1}{5}}) + 1000 \cdot - 1000 + \frac{10000 {(7 (x^{\frac{1}{5}} - 10))}^{2}}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

解题步骤 5.2.3.19

化简。

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解题步骤 5.2.3.19.1

将 $- 30$ 乘以 $1000$ 。

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + 50 x^{\frac{4}{5}} - 2000 x^{\frac{3}{5}} + 30000 x^{\frac{2}{5}} - 200000 x^{\frac{1}{5}} + 500000 + 1000 x^{\frac{3}{5}} - 30000 x^{\frac{2}{5}} + 1000 (300 x^{\frac{1}{5}}) + 1000 \cdot - 1000 + \frac{10000 {(7 (x^{\frac{1}{5}} - 10))}^{2}}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

解题步骤 5.2.3.19.2

将 $300$ 乘以 $1000$ 。

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + 50 x^{\frac{4}{5}} - 2000 x^{\frac{3}{5}} + 30000 x^{\frac{2}{5}} - 200000 x^{\frac{1}{5}} + 500000 + 1000 x^{\frac{3}{5}} - 30000 x^{\frac{2}{5}} + 300000 x^{\frac{1}{5}} + 1000 \cdot - 1000 + \frac{10000 {(7 (x^{\frac{1}{5}} - 10))}^{2}}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

解题步骤 5.2.3.19.3

将 $1000$ 乘以 $- 1000$ 。

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + 50 x^{\frac{4}{5}} - 2000 x^{\frac{3}{5}} + 30000 x^{\frac{2}{5}} - 200000 x^{\frac{1}{5}} + 500000 + 1000 x^{\frac{3}{5}} - 30000 x^{\frac{2}{5}} + 300000 x^{\frac{1}{5}} - 1000000 + \frac{10000 {(7 (x^{\frac{1}{5}} - 10))}^{2}}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

解题步骤 5.2.3.20

化简分子。

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解题步骤 5.2.3.20.1

对 $7 (x^{\frac{1}{5}} - 10)$ 运用乘积法则。

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + 50 x^{\frac{4}{5}} - 2000 x^{\frac{3}{5}} + 30000 x^{\frac{2}{5}} - 200000 x^{\frac{1}{5}} + 500000 + 1000 x^{\frac{3}{5}} - 30000 x^{\frac{2}{5}} + 300000 x^{\frac{1}{5}} - 1000000 + \frac{10000 \cdot (7^{2} {(x^{\frac{1}{5}} - 10)}^{2})}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

解题步骤 5.2.3.20.2

对 $7$ 进行 $2$ 次方运算。

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + 50 x^{\frac{4}{5}} - 2000 x^{\frac{3}{5}} + 30000 x^{\frac{2}{5}} - 200000 x^{\frac{1}{5}} + 500000 + 1000 x^{\frac{3}{5}} - 30000 x^{\frac{2}{5}} + 300000 x^{\frac{1}{5}} - 1000000 + \frac{10000 \cdot (49 {(x^{\frac{1}{5}} - 10)}^{2})}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

解题步骤 5.2.3.20.3

将 $10000$ 乘以 $49$ 。

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + 50 x^{\frac{4}{5}} - 2000 x^{\frac{3}{5}} + 30000 x^{\frac{2}{5}} - 200000 x^{\frac{1}{5}} + 500000 + 1000 x^{\frac{3}{5}} - 30000 x^{\frac{2}{5}} + 300000 x^{\frac{1}{5}} - 1000000 + \frac{490000 {(x^{\frac{1}{5}} - 10)}^{2}}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

解题步骤 5.2.3.21

从 $490000 {(x^{\frac{1}{5}} - 10)}^{2}$ 中分解出因数 $49$ 。

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + 50 x^{\frac{4}{5}} - 2000 x^{\frac{3}{5}} + 30000 x^{\frac{2}{5}} - 200000 x^{\frac{1}{5}} + 500000 + 1000 x^{\frac{3}{5}} - 30000 x^{\frac{2}{5}} + 300000 x^{\frac{1}{5}} - 1000000 + \frac{49 (10000 {(x^{\frac{1}{5}} - 10)}^{2})}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

解题步骤 5.2.3.22

约去公因数。

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解题步骤 5.2.3.22.1

从 $49$ 中分解出因数 $49$ 。

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + 50 x^{\frac{4}{5}} - 2000 x^{\frac{3}{5}} + 30000 x^{\frac{2}{5}} - 200000 x^{\frac{1}{5}} + 500000 + 1000 x^{\frac{3}{5}} - 30000 x^{\frac{2}{5}} + 300000 x^{\frac{1}{5}} - 1000000 + \frac{49 (10000 {(x^{\frac{1}{5}} - 10)}^{2})}{49 (1)} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

解题步骤 5.2.3.22.2

约去公因数。

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + 50 x^{\frac{4}{5}} - 2000 x^{\frac{3}{5}} + 30000 x^{\frac{2}{5}} - 200000 x^{\frac{1}{5}} + 500000 + 1000 x^{\frac{3}{5}} - 30000 x^{\frac{2}{5}} + 300000 x^{\frac{1}{5}} - 1000000 + \frac{49 (10000 {(x^{\frac{1}{5}} - 10)}^{2})}{49 \cdot 1} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

解题步骤 5.2.3.22.3

重写表达式。

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + 50 x^{\frac{4}{5}} - 2000 x^{\frac{3}{5}} + 30000 x^{\frac{2}{5}} - 200000 x^{\frac{1}{5}} + 500000 + 1000 x^{\frac{3}{5}} - 30000 x^{\frac{2}{5}} + 300000 x^{\frac{1}{5}} - 1000000 + \frac{10000 {(x^{\frac{1}{5}} - 10)}^{2}}{1} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

解题步骤 5.2.3.22.4

用 $10000 {(x^{\frac{1}{5}} - 10)}^{2}$ 除以 $1$ 。

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + 50 x^{\frac{4}{5}} - 2000 x^{\frac{3}{5}} + 30000 x^{\frac{2}{5}} - 200000 x^{\frac{1}{5}} + 500000 + 1000 x^{\frac{3}{5}} - 30000 x^{\frac{2}{5}} + 300000 x^{\frac{1}{5}} - 1000000 + 10000 {(x^{\frac{1}{5}} - 10)}^{2} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

解题步骤 5.2.3.23

将 ${(x^{\frac{1}{5}} - 10)}^{2}$ 重写为 $(x^{\frac{1}{5}} - 10) (x^{\frac{1}{5}} - 10)$ 。

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + 50 x^{\frac{4}{5}} - 2000 x^{\frac{3}{5}} + 30000 x^{\frac{2}{5}} - 200000 x^{\frac{1}{5}} + 500000 + 1000 x^{\frac{3}{5}} - 30000 x^{\frac{2}{5}} + 300000 x^{\frac{1}{5}} - 1000000 + 10000 ((x^{\frac{1}{5}} - 10) (x^{\frac{1}{5}} - 10)) + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

解题步骤 5.2.3.24

使用 FOIL 方法展开 $(x^{\frac{1}{5}} - 10) (x^{\frac{1}{5}} - 10)$ 。

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解题步骤 5.2.3.24.1

运用分配律。

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + 50 x^{\frac{4}{5}} - 2000 x^{\frac{3}{5}} + 30000 x^{\frac{2}{5}} - 200000 x^{\frac{1}{5}} + 500000 + 1000 x^{\frac{3}{5}} - 30000 x^{\frac{2}{5}} + 300000 x^{\frac{1}{5}} - 1000000 + 10000 (x^{\frac{1}{5}} (x^{\frac{1}{5}} - 10) - 10 (x^{\frac{1}{5}} - 10)) + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

解题步骤 5.2.3.24.2

运用分配律。

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + 50 x^{\frac{4}{5}} - 2000 x^{\frac{3}{5}} + 30000 x^{\frac{2}{5}} - 200000 x^{\frac{1}{5}} + 500000 + 1000 x^{\frac{3}{5}} - 30000 x^{\frac{2}{5}} + 300000 x^{\frac{1}{5}} - 1000000 + 10000 (x^{\frac{1}{5}} x^{\frac{1}{5}} + x^{\frac{1}{5}} \cdot - 10 - 10 (x^{\frac{1}{5}} - 10)) + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

解题步骤 5.2.3.24.3

运用分配律。

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + 50 x^{\frac{4}{5}} - 2000 x^{\frac{3}{5}} + 30000 x^{\frac{2}{5}} - 200000 x^{\frac{1}{5}} + 500000 + 1000 x^{\frac{3}{5}} - 30000 x^{\frac{2}{5}} + 300000 x^{\frac{1}{5}} - 1000000 + 10000 (x^{\frac{1}{5}} x^{\frac{1}{5}} + x^{\frac{1}{5}} \cdot - 10 - 10 x^{\frac{1}{5}} - 10 \cdot - 10) + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

解题步骤 5.2.3.25

化简并合并同类项。

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解题步骤 5.2.3.25.1

化简每一项。

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解题步骤 5.2.3.25.1.1

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 5.2.3.25.1.1.1

使用幂法则 $a^{m} a^{n} = a^{m + n}$ 合并指数。

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + 50 x^{\frac{4}{5}} - 2000 x^{\frac{3}{5}} + 30000 x^{\frac{2}{5}} - 200000 x^{\frac{1}{5}} + 500000 + 1000 x^{\frac{3}{5}} - 30000 x^{\frac{2}{5}} + 300000 x^{\frac{1}{5}} - 1000000 + 10000 (x^{\frac{1}{5} + \frac{1}{5}} + x^{\frac{1}{5}} \cdot - 10 - 10 x^{\frac{1}{5}} - 10 \cdot - 10) + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

解题步骤 5.2.3.25.1.1.2

在公分母上合并分子。

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + 50 x^{\frac{4}{5}} - 2000 x^{\frac{3}{5}} + 30000 x^{\frac{2}{5}} - 200000 x^{\frac{1}{5}} + 500000 + 1000 x^{\frac{3}{5}} - 30000 x^{\frac{2}{5}} + 300000 x^{\frac{1}{5}} - 1000000 + 10000 (x^{\frac{1 + 1}{5}} + x^{\frac{1}{5}} \cdot - 10 - 10 x^{\frac{1}{5}} - 10 \cdot - 10) + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

解题步骤 5.2.3.25.1.1.3

将 $1$ 和 $1$ 相加。

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + 50 x^{\frac{4}{5}} - 2000 x^{\frac{3}{5}} + 30000 x^{\frac{2}{5}} - 200000 x^{\frac{1}{5}} + 500000 + 1000 x^{\frac{3}{5}} - 30000 x^{\frac{2}{5}} + 300000 x^{\frac{1}{5}} - 1000000 + 10000 (x^{\frac{2}{5}} + x^{\frac{1}{5}} \cdot - 10 - 10 x^{\frac{1}{5}} - 10 \cdot - 10) + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

解题步骤 5.2.3.25.1.2

将 $- 10$ 移到 $x^{\frac{1}{5}}$ 的左侧。

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + 50 x^{\frac{4}{5}} - 2000 x^{\frac{3}{5}} + 30000 x^{\frac{2}{5}} - 200000 x^{\frac{1}{5}} + 500000 + 1000 x^{\frac{3}{5}} - 30000 x^{\frac{2}{5}} + 300000 x^{\frac{1}{5}} - 1000000 + 10000 (x^{\frac{2}{5}} - 10 \cdot x^{\frac{1}{5}} - 10 x^{\frac{1}{5}} - 10 \cdot - 10) + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

解题步骤 5.2.3.25.1.3

将 $- 10$ 乘以 $- 10$ 。

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + 50 x^{\frac{4}{5}} - 2000 x^{\frac{3}{5}} + 30000 x^{\frac{2}{5}} - 200000 x^{\frac{1}{5}} + 500000 + 1000 x^{\frac{3}{5}} - 30000 x^{\frac{2}{5}} + 300000 x^{\frac{1}{5}} - 1000000 + 10000 (x^{\frac{2}{5}} - 10 x^{\frac{1}{5}} - 10 x^{\frac{1}{5}} + 100) + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

解题步骤 5.2.3.25.2

从 $- 10 x^{\frac{1}{5}}$ 中减去 $10 x^{\frac{1}{5}}$ 。

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + 50 x^{\frac{4}{5}} - 2000 x^{\frac{3}{5}} + 30000 x^{\frac{2}{5}} - 200000 x^{\frac{1}{5}} + 500000 + 1000 x^{\frac{3}{5}} - 30000 x^{\frac{2}{5}} + 300000 x^{\frac{1}{5}} - 1000000 + 10000 (x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 100) + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

解题步骤 5.2.3.26

运用分配律。

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + 50 x^{\frac{4}{5}} - 2000 x^{\frac{3}{5}} + 30000 x^{\frac{2}{5}} - 200000 x^{\frac{1}{5}} + 500000 + 1000 x^{\frac{3}{5}} - 30000 x^{\frac{2}{5}} + 300000 x^{\frac{1}{5}} - 1000000 + 10000 x^{\frac{2}{5}} + 10000 (- 20 x^{\frac{1}{5}}) + 10000 \cdot 100 + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

解题步骤 5.2.3.27

化简。

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解题步骤 5.2.3.27.1

将 $- 20$ 乘以 $10000$ 。

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + 50 x^{\frac{4}{5}} - 2000 x^{\frac{3}{5}} + 30000 x^{\frac{2}{5}} - 200000 x^{\frac{1}{5}} + 500000 + 1000 x^{\frac{3}{5}} - 30000 x^{\frac{2}{5}} + 300000 x^{\frac{1}{5}} - 1000000 + 10000 x^{\frac{2}{5}} - 200000 x^{\frac{1}{5}} + 10000 \cdot 100 + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

解题步骤 5.2.3.27.2

将 $10000$ 乘以 $100$ 。

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + 50 x^{\frac{4}{5}} - 2000 x^{\frac{3}{5}} + 30000 x^{\frac{2}{5}} - 200000 x^{\frac{1}{5}} + 500000 + 1000 x^{\frac{3}{5}} - 30000 x^{\frac{2}{5}} + 300000 x^{\frac{1}{5}} - 1000000 + 10000 x^{\frac{2}{5}} - 200000 x^{\frac{1}{5}} + 1000000 + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

解题步骤 5.2.3.28

约去公因数。

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + 50 x^{\frac{4}{5}} - 2000 x^{\frac{3}{5}} + 30000 x^{\frac{2}{5}} - 200000 x^{\frac{1}{5}} + 500000 + 1000 x^{\frac{3}{5}} - 30000 x^{\frac{2}{5}} + 300000 x^{\frac{1}{5}} - 1000000 + 10000 x^{\frac{2}{5}} - 200000 x^{\frac{1}{5}} + 1000000 + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

解题步骤 5.2.3.29

用 $50000 (x^{\frac{1}{5}} - 10)$ 除以 $1$ 。

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + 50 x^{\frac{4}{5}} - 2000 x^{\frac{3}{5}} + 30000 x^{\frac{2}{5}} - 200000 x^{\frac{1}{5}} + 500000 + 1000 x^{\frac{3}{5}} - 30000 x^{\frac{2}{5}} + 300000 x^{\frac{1}{5}} - 1000000 + 10000 x^{\frac{2}{5}} - 200000 x^{\frac{1}{5}} + 1000000 + 50000 (x^{\frac{1}{5}} - 10) + 100000$

解题步骤 5.2.3.30

运用分配律。

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + 50 x^{\frac{4}{5}} - 2000 x^{\frac{3}{5}} + 30000 x^{\frac{2}{5}} - 200000 x^{\frac{1}{5}} + 500000 + 1000 x^{\frac{3}{5}} - 30000 x^{\frac{2}{5}} + 300000 x^{\frac{1}{5}} - 1000000 + 10000 x^{\frac{2}{5}} - 200000 x^{\frac{1}{5}} + 1000000 + 50000 x^{\frac{1}{5}} + 50000 \cdot - 10 + 100000$

解题步骤 5.2.3.31

将 $50000$ 乘以 $- 10$ 。

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + 50 x^{\frac{4}{5}} - 2000 x^{\frac{3}{5}} + 30000 x^{\frac{2}{5}} - 200000 x^{\frac{1}{5}} + 500000 + 1000 x^{\frac{3}{5}} - 30000 x^{\frac{2}{5}} + 300000 x^{\frac{1}{5}} - 1000000 + 10000 x^{\frac{2}{5}} - 200000 x^{\frac{1}{5}} + 1000000 + 50000 x^{\frac{1}{5}} - 500000 + 100000$

解题步骤 5.2.4

通过加上各项进行化简。

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解题步骤 5.2.4.1

合并 $x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + 50 x^{\frac{4}{5}} - 2000 x^{\frac{3}{5}} + 30000 x^{\frac{2}{5}} - 200000 x^{\frac{1}{5}} + 500000 + 1000 x^{\frac{3}{5}} - 30000 x^{\frac{2}{5}} + 300000 x^{\frac{1}{5}} - 1000000 + 10000 x^{\frac{2}{5}} - 200000 x^{\frac{1}{5}} + 1000000 + 50000 x^{\frac{1}{5}} - 500000 + 100000$ 中相反的项。

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解题步骤 5.2.4.1.1

将 $- 50 x^{\frac{4}{5}}$ 和 $50 x^{\frac{4}{5}}$ 相加。

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + 0 - 2000 x^{\frac{3}{5}} + 30000 x^{\frac{2}{5}} - 200000 x^{\frac{1}{5}} + 500000 + 1000 x^{\frac{3}{5}} - 30000 x^{\frac{2}{5}} + 300000 x^{\frac{1}{5}} - 1000000 + 10000 x^{\frac{2}{5}} - 200000 x^{\frac{1}{5}} + 1000000 + 50000 x^{\frac{1}{5}} - 500000 + 100000$

解题步骤 5.2.4.1.2

将 $x + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000$ 和 $0$ 相加。

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 - 2000 x^{\frac{3}{5}} + 30000 x^{\frac{2}{5}} - 200000 x^{\frac{1}{5}} + 500000 + 1000 x^{\frac{3}{5}} - 30000 x^{\frac{2}{5}} + 300000 x^{\frac{1}{5}} - 1000000 + 10000 x^{\frac{2}{5}} - 200000 x^{\frac{1}{5}} + 1000000 + 50000 x^{\frac{1}{5}} - 500000 + 100000$

解题步骤 5.2.4.1.3

从 $30000 x^{\frac{2}{5}}$ 中减去 $30000 x^{\frac{2}{5}}$ 。

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 - 2000 x^{\frac{3}{5}} - 200000 x^{\frac{1}{5}} + 500000 + 1000 x^{\frac{3}{5}} + 0 + 300000 x^{\frac{1}{5}} - 1000000 + 10000 x^{\frac{2}{5}} - 200000 x^{\frac{1}{5}} + 1000000 + 50000 x^{\frac{1}{5}} - 500000 + 100000$

解题步骤 5.2.4.1.4

将 $x + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 - 2000 x^{\frac{3}{5}} - 200000 x^{\frac{1}{5}} + 500000 + 1000 x^{\frac{3}{5}}$ 和 $0$ 相加。

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 - 2000 x^{\frac{3}{5}} - 200000 x^{\frac{1}{5}} + 500000 + 1000 x^{\frac{3}{5}} + 300000 x^{\frac{1}{5}} - 1000000 + 10000 x^{\frac{2}{5}} - 200000 x^{\frac{1}{5}} + 1000000 + 50000 x^{\frac{1}{5}} - 500000 + 100000$

解题步骤 5.2.4.1.5

将 $- 10000 x^{\frac{2}{5}}$ 和 $10000 x^{\frac{2}{5}}$ 相加。

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x + 1000 x^{\frac{3}{5}} + 50000 x^{\frac{1}{5}} - 100000 - 2000 x^{\frac{3}{5}} - 200000 x^{\frac{1}{5}} + 500000 + 1000 x^{\frac{3}{5}} + 300000 x^{\frac{1}{5}} - 1000000 + 0 - 200000 x^{\frac{1}{5}} + 1000000 + 50000 x^{\frac{1}{5}} - 500000 + 100000$

解题步骤 5.2.4.1.6

将 $x + 1000 x^{\frac{3}{5}} + 50000 x^{\frac{1}{5}} - 100000 - 2000 x^{\frac{3}{5}} - 200000 x^{\frac{1}{5}} + 500000 + 1000 x^{\frac{3}{5}} + 300000 x^{\frac{1}{5}} - 1000000$ 和 $0$ 相加。

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x + 1000 x^{\frac{3}{5}} + 50000 x^{\frac{1}{5}} - 100000 - 2000 x^{\frac{3}{5}} - 200000 x^{\frac{1}{5}} + 500000 + 1000 x^{\frac{3}{5}} + 300000 x^{\frac{1}{5}} - 1000000 - 200000 x^{\frac{1}{5}} + 1000000 + 50000 x^{\frac{1}{5}} - 500000 + 100000$

解题步骤 5.2.4.1.7

将 $- 1000000$ 和 $1000000$ 相加。

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x + 1000 x^{\frac{3}{5}} + 50000 x^{\frac{1}{5}} - 100000 - 2000 x^{\frac{3}{5}} - 200000 x^{\frac{1}{5}} + 500000 + 1000 x^{\frac{3}{5}} + 300000 x^{\frac{1}{5}} - 200000 x^{\frac{1}{5}} + 0 + 50000 x^{\frac{1}{5}} - 500000 + 100000$

解题步骤 5.2.4.1.8

将 $x + 1000 x^{\frac{3}{5}} + 50000 x^{\frac{1}{5}} - 100000 - 2000 x^{\frac{3}{5}} - 200000 x^{\frac{1}{5}} + 500000 + 1000 x^{\frac{3}{5}} + 300000 x^{\frac{1}{5}} - 200000 x^{\frac{1}{5}}$ 和 $0$ 相加。

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x + 1000 x^{\frac{3}{5}} + 50000 x^{\frac{1}{5}} - 100000 - 2000 x^{\frac{3}{5}} - 200000 x^{\frac{1}{5}} + 500000 + 1000 x^{\frac{3}{5}} + 300000 x^{\frac{1}{5}} - 200000 x^{\frac{1}{5}} + 50000 x^{\frac{1}{5}} - 500000 + 100000$

解题步骤 5.2.4.1.9

从 $500000$ 中减去 $500000$ 。

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x + 1000 x^{\frac{3}{5}} + 50000 x^{\frac{1}{5}} - 100000 - 2000 x^{\frac{3}{5}} - 200000 x^{\frac{1}{5}} + 1000 x^{\frac{3}{5}} + 300000 x^{\frac{1}{5}} - 200000 x^{\frac{1}{5}} + 50000 x^{\frac{1}{5}} + 0 + 100000$

解题步骤 5.2.4.1.10

将 $x + 1000 x^{\frac{3}{5}} + 50000 x^{\frac{1}{5}} - 100000 - 2000 x^{\frac{3}{5}} - 200000 x^{\frac{1}{5}} + 1000 x^{\frac{3}{5}} + 300000 x^{\frac{1}{5}} - 200000 x^{\frac{1}{5}} + 50000 x^{\frac{1}{5}}$ 和 $0$ 相加。

解题步骤 5.2.4.1.11

将 $- 100000$ 和 $100000$ 相加。

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x + 1000 x^{\frac{3}{5}} + 50000 x^{\frac{1}{5}} - 2000 x^{\frac{3}{5}} - 200000 x^{\frac{1}{5}} + 1000 x^{\frac{3}{5}} + 300000 x^{\frac{1}{5}} - 200000 x^{\frac{1}{5}} + 50000 x^{\frac{1}{5}} + 0$

解题步骤 5.2.4.1.12

将 $x + 1000 x^{\frac{3}{5}} + 50000 x^{\frac{1}{5}} - 2000 x^{\frac{3}{5}} - 200000 x^{\frac{1}{5}} + 1000 x^{\frac{3}{5}} + 300000 x^{\frac{1}{5}} - 200000 x^{\frac{1}{5}} + 50000 x^{\frac{1}{5}}$ 和 $0$ 相加。

解题步骤 5.2.4.2

从 $1000 x^{\frac{3}{5}}$ 中减去 $2000 x^{\frac{3}{5}}$ 。

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x + 50000 x^{\frac{1}{5}} - 1000 x^{\frac{3}{5}} - 200000 x^{\frac{1}{5}} + 1000 x^{\frac{3}{5}} + 300000 x^{\frac{1}{5}} - 200000 x^{\frac{1}{5}} + 50000 x^{\frac{1}{5}}$

解题步骤 5.2.4.3

合并 $x + 50000 x^{\frac{1}{5}} - 1000 x^{\frac{3}{5}} - 200000 x^{\frac{1}{5}} + 1000 x^{\frac{3}{5}} + 300000 x^{\frac{1}{5}} - 200000 x^{\frac{1}{5}} + 50000 x^{\frac{1}{5}}$ 中相反的项。

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解题步骤 5.2.4.3.1

将 $- 1000 x^{\frac{3}{5}}$ 和 $1000 x^{\frac{3}{5}}$ 相加。

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x + 50000 x^{\frac{1}{5}} - 200000 x^{\frac{1}{5}} + 0 + 300000 x^{\frac{1}{5}} - 200000 x^{\frac{1}{5}} + 50000 x^{\frac{1}{5}}$

解题步骤 5.2.4.3.2

将 $x + 50000 x^{\frac{1}{5}} - 200000 x^{\frac{1}{5}}$ 和 $0$ 相加。

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x + 50000 x^{\frac{1}{5}} - 200000 x^{\frac{1}{5}} + 300000 x^{\frac{1}{5}} - 200000 x^{\frac{1}{5}} + 50000 x^{\frac{1}{5}}$

解题步骤 5.2.4.4

从 $50000 x^{\frac{1}{5}}$ 中减去 $200000 x^{\frac{1}{5}}$ 。

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 150000 x^{\frac{1}{5}} + 300000 x^{\frac{1}{5}} - 200000 x^{\frac{1}{5}} + 50000 x^{\frac{1}{5}}$

解题步骤 5.2.4.5

将 $- 150000 x^{\frac{1}{5}}$ 和 $300000 x^{\frac{1}{5}}$ 相加。

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x + 150000 x^{\frac{1}{5}} - 200000 x^{\frac{1}{5}} + 50000 x^{\frac{1}{5}}$

解题步骤 5.2.4.6

从 $150000 x^{\frac{1}{5}}$ 中减去 $200000 x^{\frac{1}{5}}$ 。

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50000 x^{\frac{1}{5}} + 50000 x^{\frac{1}{5}}$

解题步骤 5.2.4.7

合并 $x - 50000 x^{\frac{1}{5}} + 50000 x^{\frac{1}{5}}$ 中相反的项。

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解题步骤 5.2.4.7.1

将 $- 50000 x^{\frac{1}{5}}$ 和 $50000 x^{\frac{1}{5}}$ 相加。

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x + 0$

解题步骤 5.2.4.7.2

将 $x$ 和 $0$ 相加。

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x$

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 $f (\frac{x^{5}}{16807} + \frac{50 x^{4}}{2401} + \frac{1000 x^{3}}{343} + \frac{10000 x^{2}}{49} + \frac{50000 x}{7} + 100000)$ 。

$f (\frac{x^{5}}{16807} + \frac{50 x^{4}}{2401} + \frac{1000 x^{3}}{343} + \frac{10000 x^{2}}{49} + \frac{50000 x}{7} + 100000) = 7 ({(\frac{x^{5}}{16807} + \frac{50 x^{4}}{2401} + \frac{1000 x^{3}}{343} + \frac{10000 x^{2}}{49} + \frac{50000 x}{7} + 100000)}^{\frac{1}{5}} - 10)$

解题步骤 5.3.3

运用分配律。

$f (\frac{x^{5}}{16807} + \frac{50 x^{4}}{2401} + \frac{1000 x^{3}}{343} + \frac{10000 x^{2}}{49} + \frac{50000 x}{7} + 100000) = 7 {(\frac{x^{5}}{16807} + \frac{50 x^{4}}{2401} + \frac{1000 x^{3}}{343} + \frac{10000 x^{2}}{49} + \frac{50000 x}{7} + 100000)}^{\frac{1}{5}} + 7 \cdot - 10$

解题步骤 5.3.4

将 $7$ 乘以 $- 10$ 。

解题步骤 5.4

由于 $f^{- 1} (f (x)) = x$ 和 $f (f^{- 1} (x)) = x$ ，因此 $f^{- 1} (x) = \frac{x^{5}}{16807} + \frac{50 x^{4}}{2401} + \frac{1000 x^{3}}{343} + \frac{10000 x^{2}}{49} + \frac{50000 x}{7} + 100000$ 为 $f (x) = 7 (x^{\frac{1}{5}} - 10)$ 的反函数。

$f^{- 1} (x) = \frac{x^{5}}{16807} + \frac{50 x^{4}}{2401} + \frac{1000 x^{3}}{343} + \frac{10000 x^{2}}{49} + \frac{50000 x}{7} + 100000$

初级微积分 示例

初级微积分示例