微积分学示例

${(\cos (\frac{3 π}{12}) + i \sin (\frac{3 π}{12}))}^{5}$

解题步骤 1

化简每一项。

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解题步骤 1.1

约去 $3$ 和 $12$ 的公因数。

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解题步骤 1.1.1

从 $3 π$ 中分解出因数 $3$ 。

${(\cos (\frac{3 (π)}{12}) + i \sin (\frac{3 π}{12}))}^{5}$

解题步骤 1.1.2

约去公因数。

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解题步骤 1.1.2.1

从 $12$ 中分解出因数 $3$ 。

${(\cos (\frac{3 π}{3 \cdot 4}) + i \sin (\frac{3 π}{12}))}^{5}$

解题步骤 1.1.2.2

约去公因数。

${(\cos (\frac{3 π}{3 \cdot 4}) + i \sin (\frac{3 π}{12}))}^{5}$

解题步骤 1.1.2.3

重写表达式。

${(\cos (\frac{π}{4}) + i \sin (\frac{3 π}{12}))}^{5}$

解题步骤 1.2

$\cos (\frac{π}{4})$ 的准确值为 $\frac{\sqrt{2}}{2}$ 。

${(\frac{\sqrt{2}}{2} + i \sin (\frac{3 π}{12}))}^{5}$

解题步骤 1.3

约去 $3$ 和 $12$ 的公因数。

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解题步骤 1.3.1

从 $3 π$ 中分解出因数 $3$ 。

${(\frac{\sqrt{2}}{2} + i \sin (\frac{3 (π)}{12}))}^{5}$

解题步骤 1.3.2

约去公因数。

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解题步骤 1.3.2.1

从 $12$ 中分解出因数 $3$ 。

${(\frac{\sqrt{2}}{2} + i \sin (\frac{3 π}{3 \cdot 4}))}^{5}$

解题步骤 1.3.2.2

约去公因数。

${(\frac{\sqrt{2}}{2} + i \sin (\frac{3 π}{3 \cdot 4}))}^{5}$

解题步骤 1.3.2.3

重写表达式。

${(\frac{\sqrt{2}}{2} + i \sin (\frac{π}{4}))}^{5}$

解题步骤 1.4

$\sin (\frac{π}{4})$ 的准确值为 $\frac{\sqrt{2}}{2}$ 。

${(\frac{\sqrt{2}}{2} + i \frac{\sqrt{2}}{2})}^{5}$

解题步骤 1.5

组合 $i$ 和 $\frac{\sqrt{2}}{2}$ 。

${(\frac{\sqrt{2}}{2} + \frac{i \sqrt{2}}{2})}^{5}$

解题步骤 2

使用二项式定理。

${(\frac{\sqrt{2}}{2})}^{5} + 5 {(\frac{\sqrt{2}}{2})}^{4} \frac{i \sqrt{2}}{2} + 10 {(\frac{\sqrt{2}}{2})}^{3} {(\frac{i \sqrt{2}}{2})}^{2} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

解题步骤 3

化简项。

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解题步骤 3.1

化简每一项。

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解题步骤 3.1.1

对 $\frac{\sqrt{2}}{2}$ 运用乘积法则。

$\frac{{\sqrt{2}}^{5}}{2^{5}} + 5 {(\frac{\sqrt{2}}{2})}^{4} \frac{i \sqrt{2}}{2} + 10 {(\frac{\sqrt{2}}{2})}^{3} {(\frac{i \sqrt{2}}{2})}^{2} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

解题步骤 3.1.2

化简分子。

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解题步骤 3.1.2.1

将 ${\sqrt{2}}^{5}$ 重写为 $\sqrt{2^{5}}$ 。

$\frac{\sqrt{2^{5}}}{2^{5}} + 5 {(\frac{\sqrt{2}}{2})}^{4} \frac{i \sqrt{2}}{2} + 10 {(\frac{\sqrt{2}}{2})}^{3} {(\frac{i \sqrt{2}}{2})}^{2} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

解题步骤 3.1.2.2

对 $2$ 进行 $5$ 次方运算。

$\frac{\sqrt{32}}{2^{5}} + 5 {(\frac{\sqrt{2}}{2})}^{4} \frac{i \sqrt{2}}{2} + 10 {(\frac{\sqrt{2}}{2})}^{3} {(\frac{i \sqrt{2}}{2})}^{2} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

解题步骤 3.1.2.3

将 $32$ 重写为 $4^{2} \cdot 2$ 。

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解题步骤 3.1.2.3.1

从 $32$ 中分解出因数 $16$ 。

$\frac{\sqrt{16 (2)}}{2^{5}} + 5 {(\frac{\sqrt{2}}{2})}^{4} \frac{i \sqrt{2}}{2} + 10 {(\frac{\sqrt{2}}{2})}^{3} {(\frac{i \sqrt{2}}{2})}^{2} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

解题步骤 3.1.2.3.2

将 $16$ 重写为 $4^{2}$ 。

$\frac{\sqrt{4^{2} \cdot 2}}{2^{5}} + 5 {(\frac{\sqrt{2}}{2})}^{4} \frac{i \sqrt{2}}{2} + 10 {(\frac{\sqrt{2}}{2})}^{3} {(\frac{i \sqrt{2}}{2})}^{2} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

解题步骤 3.1.2.4

从根式下提出各项。

$\frac{4 \sqrt{2}}{2^{5}} + 5 {(\frac{\sqrt{2}}{2})}^{4} \frac{i \sqrt{2}}{2} + 10 {(\frac{\sqrt{2}}{2})}^{3} {(\frac{i \sqrt{2}}{2})}^{2} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

解题步骤 3.1.3

对 $2$ 进行 $5$ 次方运算。

$\frac{4 \sqrt{2}}{32} + 5 {(\frac{\sqrt{2}}{2})}^{4} \frac{i \sqrt{2}}{2} + 10 {(\frac{\sqrt{2}}{2})}^{3} {(\frac{i \sqrt{2}}{2})}^{2} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

解题步骤 3.1.4

约去 $4$ 和 $32$ 的公因数。

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解题步骤 3.1.4.1

从 $4 \sqrt{2}$ 中分解出因数 $4$ 。

$\frac{4 (\sqrt{2})}{32} + 5 {(\frac{\sqrt{2}}{2})}^{4} \frac{i \sqrt{2}}{2} + 10 {(\frac{\sqrt{2}}{2})}^{3} {(\frac{i \sqrt{2}}{2})}^{2} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

解题步骤 3.1.4.2

约去公因数。

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解题步骤 3.1.4.2.1

从 $32$ 中分解出因数 $4$ 。

$\frac{4 \sqrt{2}}{4 \cdot 8} + 5 {(\frac{\sqrt{2}}{2})}^{4} \frac{i \sqrt{2}}{2} + 10 {(\frac{\sqrt{2}}{2})}^{3} {(\frac{i \sqrt{2}}{2})}^{2} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

解题步骤 3.1.4.2.2

约去公因数。

解题步骤 3.1.4.2.3

重写表达式。

$\frac{\sqrt{2}}{8} + 5 {(\frac{\sqrt{2}}{2})}^{4} \frac{i \sqrt{2}}{2} + 10 {(\frac{\sqrt{2}}{2})}^{3} {(\frac{i \sqrt{2}}{2})}^{2} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

解题步骤 3.1.5

对 $\frac{\sqrt{2}}{2}$ 运用乘积法则。

$\frac{\sqrt{2}}{8} + 5 \frac{{\sqrt{2}}^{4}}{2^{4}} \frac{i \sqrt{2}}{2} + 10 {(\frac{\sqrt{2}}{2})}^{3} {(\frac{i \sqrt{2}}{2})}^{2} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

解题步骤 3.1.6

化简分子。

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解题步骤 3.1.6.1

将 ${\sqrt{2}}^{4}$ 重写为 $2^{2}$ 。

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解题步骤 3.1.6.1.1

使用 $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ ，将 $\sqrt{2}$ 重写成 $2^{\frac{1}{2}}$ 。

$\frac{\sqrt{2}}{8} + 5 \frac{{(2^{\frac{1}{2}})}^{4}}{2^{4}} \frac{i \sqrt{2}}{2} + 10 {(\frac{\sqrt{2}}{2})}^{3} {(\frac{i \sqrt{2}}{2})}^{2} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

解题步骤 3.1.6.1.2

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

$\frac{\sqrt{2}}{8} + 5 \frac{2^{\frac{1}{2} \cdot 4}}{2^{4}} \frac{i \sqrt{2}}{2} + 10 {(\frac{\sqrt{2}}{2})}^{3} {(\frac{i \sqrt{2}}{2})}^{2} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

解题步骤 3.1.6.1.3

组合 $\frac{1}{2}$ 和 $4$ 。

$\frac{\sqrt{2}}{8} + 5 \frac{2^{\frac{4}{2}}}{2^{4}} \frac{i \sqrt{2}}{2} + 10 {(\frac{\sqrt{2}}{2})}^{3} {(\frac{i \sqrt{2}}{2})}^{2} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

解题步骤 3.1.6.1.4

约去 $4$ 和 $2$ 的公因数。

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解题步骤 3.1.6.1.4.1

从 $4$ 中分解出因数 $2$ 。

$\frac{\sqrt{2}}{8} + 5 \frac{2^{\frac{2 \cdot 2}{2}}}{2^{4}} \frac{i \sqrt{2}}{2} + 10 {(\frac{\sqrt{2}}{2})}^{3} {(\frac{i \sqrt{2}}{2})}^{2} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

解题步骤 3.1.6.1.4.2

约去公因数。

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解题步骤 3.1.6.1.4.2.1

从 $2$ 中分解出因数 $2$ 。

$\frac{\sqrt{2}}{8} + 5 \frac{2^{\frac{2 \cdot 2}{2 (1)}}}{2^{4}} \frac{i \sqrt{2}}{2} + 10 {(\frac{\sqrt{2}}{2})}^{3} {(\frac{i \sqrt{2}}{2})}^{2} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

解题步骤 3.1.6.1.4.2.2

约去公因数。

$\frac{\sqrt{2}}{8} + 5 \frac{2^{\frac{2 \cdot 2}{2 \cdot 1}}}{2^{4}} \frac{i \sqrt{2}}{2} + 10 {(\frac{\sqrt{2}}{2})}^{3} {(\frac{i \sqrt{2}}{2})}^{2} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

解题步骤 3.1.6.1.4.2.3

重写表达式。

$\frac{\sqrt{2}}{8} + 5 \frac{2^{\frac{2}{1}}}{2^{4}} \frac{i \sqrt{2}}{2} + 10 {(\frac{\sqrt{2}}{2})}^{3} {(\frac{i \sqrt{2}}{2})}^{2} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

解题步骤 3.1.6.1.4.2.4

用 $2$ 除以 $1$ 。

$\frac{\sqrt{2}}{8} + 5 \frac{2^{2}}{2^{4}} \frac{i \sqrt{2}}{2} + 10 {(\frac{\sqrt{2}}{2})}^{3} {(\frac{i \sqrt{2}}{2})}^{2} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

解题步骤 3.1.6.2

对 $2$ 进行 $2$ 次方运算。

$\frac{\sqrt{2}}{8} + 5 \frac{4}{2^{4}} \frac{i \sqrt{2}}{2} + 10 {(\frac{\sqrt{2}}{2})}^{3} {(\frac{i \sqrt{2}}{2})}^{2} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

解题步骤 3.1.7

对 $2$ 进行 $4$ 次方运算。

$\frac{\sqrt{2}}{8} + 5 (\frac{4}{16}) \frac{i \sqrt{2}}{2} + 10 {(\frac{\sqrt{2}}{2})}^{3} {(\frac{i \sqrt{2}}{2})}^{2} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

解题步骤 3.1.8

约去 $4$ 和 $16$ 的公因数。

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解题步骤 3.1.8.1

从 $4$ 中分解出因数 $4$ 。

$\frac{\sqrt{2}}{8} + 5 \frac{4 (1)}{16} \frac{i \sqrt{2}}{2} + 10 {(\frac{\sqrt{2}}{2})}^{3} {(\frac{i \sqrt{2}}{2})}^{2} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

解题步骤 3.1.8.2

约去公因数。

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解题步骤 3.1.8.2.1

从 $16$ 中分解出因数 $4$ 。

$\frac{\sqrt{2}}{8} + 5 \frac{4 \cdot 1}{4 \cdot 4} \frac{i \sqrt{2}}{2} + 10 {(\frac{\sqrt{2}}{2})}^{3} {(\frac{i \sqrt{2}}{2})}^{2} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

解题步骤 3.1.8.2.2

约去公因数。

解题步骤 3.1.8.2.3

重写表达式。

$\frac{\sqrt{2}}{8} + 5 (\frac{1}{4}) \frac{i \sqrt{2}}{2} + 10 {(\frac{\sqrt{2}}{2})}^{3} {(\frac{i \sqrt{2}}{2})}^{2} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

解题步骤 3.1.9

组合 $5$ 和 $\frac{1}{4}$ 。

$\frac{\sqrt{2}}{8} + \frac{5}{4} \cdot \frac{i \sqrt{2}}{2} + 10 {(\frac{\sqrt{2}}{2})}^{3} {(\frac{i \sqrt{2}}{2})}^{2} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

解题步骤 3.1.10

乘以 $\frac{5}{4} \cdot \frac{i \sqrt{2}}{2}$ 。

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解题步骤 3.1.10.1

将 $\frac{5}{4}$ 乘以 $\frac{i \sqrt{2}}{2}$ 。

$\frac{\sqrt{2}}{8} + \frac{5 (i \sqrt{2})}{4 \cdot 2} + 10 {(\frac{\sqrt{2}}{2})}^{3} {(\frac{i \sqrt{2}}{2})}^{2} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

解题步骤 3.1.10.2

将 $4$ 乘以 $2$ 。

$\frac{\sqrt{2}}{8} + \frac{5 (i \sqrt{2})}{8} + 10 {(\frac{\sqrt{2}}{2})}^{3} {(\frac{i \sqrt{2}}{2})}^{2} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

解题步骤 3.1.11

对 $\frac{\sqrt{2}}{2}$ 运用乘积法则。

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} + 10 \frac{{\sqrt{2}}^{3}}{2^{3}} {(\frac{i \sqrt{2}}{2})}^{2} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

解题步骤 3.1.12

化简分子。

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解题步骤 3.1.12.1

将 ${\sqrt{2}}^{3}$ 重写为 $\sqrt{2^{3}}$ 。

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} + 10 \frac{\sqrt{2^{3}}}{2^{3}} {(\frac{i \sqrt{2}}{2})}^{2} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

解题步骤 3.1.12.2

对 $2$ 进行 $3$ 次方运算。

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} + 10 \frac{\sqrt{8}}{2^{3}} {(\frac{i \sqrt{2}}{2})}^{2} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

解题步骤 3.1.12.3

将 $8$ 重写为 $2^{2} \cdot 2$ 。

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解题步骤 3.1.12.3.1

从 $8$ 中分解出因数 $4$ 。

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} + 10 \frac{\sqrt{4 (2)}}{2^{3}} {(\frac{i \sqrt{2}}{2})}^{2} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

解题步骤 3.1.12.3.2

将 $4$ 重写为 $2^{2}$ 。

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} + 10 \frac{\sqrt{2^{2} \cdot 2}}{2^{3}} {(\frac{i \sqrt{2}}{2})}^{2} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

解题步骤 3.1.12.4

从根式下提出各项。

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} + 10 \frac{2 \sqrt{2}}{2^{3}} {(\frac{i \sqrt{2}}{2})}^{2} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

解题步骤 3.1.13

对 $2$ 进行 $3$ 次方运算。

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} + 10 \frac{2 \sqrt{2}}{8} {(\frac{i \sqrt{2}}{2})}^{2} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

解题步骤 3.1.14

约去 $2$ 的公因数。

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解题步骤 3.1.14.1

从 $10$ 中分解出因数 $2$ 。

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} + 2 (5) \frac{2 \sqrt{2}}{8} {(\frac{i \sqrt{2}}{2})}^{2} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

解题步骤 3.1.14.2

从 $8$ 中分解出因数 $2$ 。

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} + 2 \cdot 5 \frac{2 \sqrt{2}}{2 \cdot 4} {(\frac{i \sqrt{2}}{2})}^{2} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

解题步骤 3.1.14.3

约去公因数。

解题步骤 3.1.14.4

重写表达式。

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} + 5 \frac{2 \sqrt{2}}{4} {(\frac{i \sqrt{2}}{2})}^{2} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

解题步骤 3.1.15

组合 $5$ 和 $\frac{2 \sqrt{2}}{4}$ 。

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} + \frac{5 (2 \sqrt{2})}{4} {(\frac{i \sqrt{2}}{2})}^{2} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

解题步骤 3.1.16

将 $2$ 乘以 $5$ 。

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} + \frac{10 \sqrt{2}}{4} {(\frac{i \sqrt{2}}{2})}^{2} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

解题步骤 3.1.17

约去 $10$ 和 $4$ 的公因数。

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解题步骤 3.1.17.1

从 $10 \sqrt{2}$ 中分解出因数 $2$ 。

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} + \frac{2 (5 \sqrt{2})}{4} {(\frac{i \sqrt{2}}{2})}^{2} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

解题步骤 3.1.17.2

约去公因数。

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解题步骤 3.1.17.2.1

从 $4$ 中分解出因数 $2$ 。

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} + \frac{2 (5 \sqrt{2})}{2 (2)} {(\frac{i \sqrt{2}}{2})}^{2} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

解题步骤 3.1.17.2.2

约去公因数。

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} + \frac{2 (5 \sqrt{2})}{2 \cdot 2} {(\frac{i \sqrt{2}}{2})}^{2} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

解题步骤 3.1.17.2.3

重写表达式。

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} + \frac{5 \sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{2} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

解题步骤 3.1.18

使用幂法则 ${(a b)}^{n} = a^{n} b^{n}$ 分解指数。

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解题步骤 3.1.18.1

对 $\frac{i \sqrt{2}}{2}$ 运用乘积法则。

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} + \frac{5 \sqrt{2}}{2} \cdot \frac{{(i \sqrt{2})}^{2}}{2^{2}} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

解题步骤 3.1.18.2

对 $i \sqrt{2}$ 运用乘积法则。

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} + \frac{5 \sqrt{2}}{2} \cdot \frac{i^{2} {\sqrt{2}}^{2}}{2^{2}} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

解题步骤 3.1.19

合并。

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} + \frac{5 \sqrt{2} (i^{2} {\sqrt{2}}^{2})}{2 \cdot 2^{2}} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

解题步骤 3.1.20

通过指数相加将 $\sqrt{2}$ 乘以 ${\sqrt{2}}^{2}$ 。

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解题步骤 3.1.20.1

移动 ${\sqrt{2}}^{2}$ 。

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} + \frac{5 ({\sqrt{2}}^{2} \sqrt{2}) i^{2}}{2 \cdot 2^{2}} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

解题步骤 3.1.20.2

将 ${\sqrt{2}}^{2}$ 乘以 $\sqrt{2}$ 。

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解题步骤 3.1.20.2.1

对 $\sqrt{2}$ 进行 $1$ 次方运算。

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} + \frac{5 ({\sqrt{2}}^{2} {\sqrt{2}}^{1}) i^{2}}{2 \cdot 2^{2}} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

解题步骤 3.1.20.2.2

使用幂法则 $a^{m} a^{n} = a^{m + n}$ 合并指数。

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} + \frac{5 {\sqrt{2}}^{2 + 1} i^{2}}{2 \cdot 2^{2}} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

解题步骤 3.1.20.3

将 $2$ 和 $1$ 相加。

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} + \frac{5 {\sqrt{2}}^{3} i^{2}}{2 \cdot 2^{2}} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

解题步骤 3.1.21

通过指数相加将 $2$ 乘以 $2^{2}$ 。

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解题步骤 3.1.21.1

将 $2$ 乘以 $2^{2}$ 。

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解题步骤 3.1.21.1.1

对 $2$ 进行 $1$ 次方运算。

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} + \frac{5 {\sqrt{2}}^{3} i^{2}}{2^{1} \cdot 2^{2}} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

解题步骤 3.1.21.1.2

使用幂法则 $a^{m} a^{n} = a^{m + n}$ 合并指数。

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} + \frac{5 {\sqrt{2}}^{3} i^{2}}{2^{1 + 2}} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

解题步骤 3.1.21.2

将 $1$ 和 $2$ 相加。

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} + \frac{5 {\sqrt{2}}^{3} i^{2}}{2^{3}} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

解题步骤 3.1.22

化简分子。

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解题步骤 3.1.22.1

将 ${\sqrt{2}}^{3}$ 重写为 $\sqrt{2^{3}}$ 。

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} + \frac{5 \sqrt{2^{3}} i^{2}}{2^{3}} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

解题步骤 3.1.22.2

对 $2$ 进行 $3$ 次方运算。

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} + \frac{5 \sqrt{8} i^{2}}{2^{3}} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

解题步骤 3.1.22.3

将 $8$ 重写为 $2^{2} \cdot 2$ 。

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解题步骤 3.1.22.3.1

从 $8$ 中分解出因数 $4$ 。

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} + \frac{5 \sqrt{4 (2)} i^{2}}{2^{3}} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

解题步骤 3.1.22.3.2

将 $4$ 重写为 $2^{2}$ 。

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} + \frac{5 \sqrt{2^{2} \cdot 2} i^{2}}{2^{3}} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

解题步骤 3.1.22.4

从根式下提出各项。

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} + \frac{5 (2 \sqrt{2}) i^{2}}{2^{3}} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

解题步骤 3.1.22.5

将 $i^{2}$ 重写为 $- 1$ 。

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} + \frac{5 \cdot 2 \sqrt{2} \cdot - 1}{2^{3}} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

解题步骤 3.1.22.6

合并指数。

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解题步骤 3.1.22.6.1

将 $5$ 乘以 $2$ 。

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} + \frac{10 \sqrt{2} \cdot - 1}{2^{3}} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

解题步骤 3.1.22.6.2

将 $- 1$ 乘以 $10$ 。

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} + \frac{- 10 \sqrt{2}}{2^{3}} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

解题步骤 3.1.23

对 $2$ 进行 $3$ 次方运算。

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} + \frac{- 10 \sqrt{2}}{8} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

解题步骤 3.1.24

约去 $- 10$ 和 $8$ 的公因数。

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解题步骤 3.1.24.1

从 $- 10 \sqrt{2}$ 中分解出因数 $2$ 。

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} + \frac{2 (- 5 \sqrt{2})}{8} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

解题步骤 3.1.24.2

约去公因数。

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解题步骤 3.1.24.2.1

从 $8$ 中分解出因数 $2$ 。

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} + \frac{2 (- 5 \sqrt{2})}{2 (4)} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

解题步骤 3.1.24.2.2

约去公因数。

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} + \frac{2 (- 5 \sqrt{2})}{2 \cdot 4} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

解题步骤 3.1.24.2.3

重写表达式。

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} + \frac{- 5 \sqrt{2}}{4} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

解题步骤 3.1.25

将负号移到分数的前面。

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{(5) \sqrt{2}}{4} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

解题步骤 3.1.26

对 $\frac{\sqrt{2}}{2}$ 运用乘积法则。

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} + 10 \frac{{\sqrt{2}}^{2}}{2^{2}} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

解题步骤 3.1.27

将 ${\sqrt{2}}^{2}$ 重写为 $2$ 。

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解题步骤 3.1.27.1

使用 $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ ，将 $\sqrt{2}$ 重写成 $2^{\frac{1}{2}}$ 。

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} + 10 \frac{{(2^{\frac{1}{2}})}^{2}}{2^{2}} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

解题步骤 3.1.27.2

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

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} + 10 \frac{2^{\frac{1}{2} \cdot 2}}{2^{2}} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

解题步骤 3.1.27.3

组合 $\frac{1}{2}$ 和 $2$ 。

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} + 10 \frac{2^{\frac{2}{2}}}{2^{2}} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

解题步骤 3.1.27.4

约去 $2$ 的公因数。

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解题步骤 3.1.27.4.1

约去公因数。

解题步骤 3.1.27.4.2

重写表达式。

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} + 10 \frac{2^{1}}{2^{2}} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

解题步骤 3.1.27.5

计算指数。

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} + 10 \frac{2}{2^{2}} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

解题步骤 3.1.28

对 $2$ 进行 $2$ 次方运算。

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} + 10 (\frac{2}{4}) {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

解题步骤 3.1.29

约去 $2$ 的公因数。

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解题步骤 3.1.29.1

从 $10$ 中分解出因数 $2$ 。

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} + 2 (5) \frac{2}{4} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

解题步骤 3.1.29.2

从 $4$ 中分解出因数 $2$ 。

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} + 2 \cdot 5 \frac{2}{2 \cdot 2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

解题步骤 3.1.29.3

约去公因数。

解题步骤 3.1.29.4

重写表达式。

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} + 5 (\frac{2}{2}) {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

解题步骤 3.1.30

组合 $5$ 和 $\frac{2}{2}$ 。

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} + \frac{5 \cdot 2}{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

解题步骤 3.1.31

将 $5$ 乘以 $2$ 。

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} + \frac{10}{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

解题步骤 3.1.32

用 $10$ 除以 $2$ 。

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} + 5 {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

解题步骤 3.1.33

使用幂法则 ${(a b)}^{n} = a^{n} b^{n}$ 分解指数。

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解题步骤 3.1.33.1

对 $\frac{i \sqrt{2}}{2}$ 运用乘积法则。

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} + 5 \frac{{(i \sqrt{2})}^{3}}{2^{3}} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

解题步骤 3.1.33.2

对 $i \sqrt{2}$ 运用乘积法则。

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} + 5 \frac{i^{3} {\sqrt{2}}^{3}}{2^{3}} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

解题步骤 3.1.34

化简分子。

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解题步骤 3.1.34.1

因式分解出 $i^{2}$ 。

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} + 5 \frac{i^{2} \cdot i {\sqrt{2}}^{3}}{2^{3}} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

解题步骤 3.1.34.2

将 $i^{2}$ 重写为 $- 1$ 。

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} + 5 \frac{- 1 \cdot i {\sqrt{2}}^{3}}{2^{3}} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

解题步骤 3.1.34.3

将 $- 1 i$ 重写为 $- i$ 。

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} + 5 \frac{- i {\sqrt{2}}^{3}}{2^{3}} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

解题步骤 3.1.34.4

将 ${\sqrt{2}}^{3}$ 重写为 $\sqrt{2^{3}}$ 。

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} + 5 \frac{- i \sqrt{2^{3}}}{2^{3}} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

解题步骤 3.1.34.5

对 $2$ 进行 $3$ 次方运算。

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} + 5 \frac{- i \sqrt{8}}{2^{3}} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

解题步骤 3.1.34.6

将 $8$ 重写为 $2^{2} \cdot 2$ 。

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解题步骤 3.1.34.6.1

从 $8$ 中分解出因数 $4$ 。

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} + 5 \frac{- i \sqrt{4 (2)}}{2^{3}} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

解题步骤 3.1.34.6.2

将 $4$ 重写为 $2^{2}$ 。

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} + 5 \frac{- i \sqrt{2^{2} \cdot 2}}{2^{3}} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

解题步骤 3.1.34.7

从根式下提出各项。

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} + 5 \frac{- i \cdot 2 \sqrt{2}}{2^{3}} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

解题步骤 3.1.34.8

将 $2$ 乘以 $- 1$ 。

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} + 5 \frac{- 2 i \sqrt{2}}{2^{3}} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

解题步骤 3.1.35

对 $2$ 进行 $3$ 次方运算。

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} + 5 \frac{- 2 i \sqrt{2}}{8} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

解题步骤 3.1.36

约去 $- 2$ 和 $8$ 的公因数。

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解题步骤 3.1.36.1

从 $- 2 i \sqrt{2}$ 中分解出因数 $2$ 。

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} + 5 \frac{2 (- i \sqrt{2})}{8} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

解题步骤 3.1.36.2

约去公因数。

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解题步骤 3.1.36.2.1

从 $8$ 中分解出因数 $2$ 。

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} + 5 \frac{2 (- i \sqrt{2})}{2 (4)} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

解题步骤 3.1.36.2.2

约去公因数。

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} + 5 \frac{2 (- i \sqrt{2})}{2 \cdot 4} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

解题步骤 3.1.36.2.3

重写表达式。

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} + 5 \frac{- i \sqrt{2}}{4} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

解题步骤 3.1.37

将负号移到分数的前面。

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} + 5 (- \frac{i \sqrt{2}}{4}) + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

解题步骤 3.1.38

乘以 $5 (- \frac{i \sqrt{2}}{4})$ 。

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解题步骤 3.1.38.1

将 $- 1$ 乘以 $5$ 。

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - 5 \frac{i \sqrt{2}}{4} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

解题步骤 3.1.38.2

组合 $- 5$ 和 $\frac{i \sqrt{2}}{4}$ 。

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} + \frac{- 5 (i \sqrt{2})}{4} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

解题步骤 3.1.39

将负号移到分数的前面。

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{(5) i \sqrt{2}}{4} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

解题步骤 3.1.40

组合 $5$ 和 $\frac{\sqrt{2}}{2}$ 。

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{5 i \sqrt{2}}{4} + \frac{5 \sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

解题步骤 3.1.41

使用幂法则 ${(a b)}^{n} = a^{n} b^{n}$ 分解指数。

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解题步骤 3.1.41.1

对 $\frac{i \sqrt{2}}{2}$ 运用乘积法则。

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{5 i \sqrt{2}}{4} + \frac{5 \sqrt{2}}{2} \cdot \frac{{(i \sqrt{2})}^{4}}{2^{4}} + {(\frac{i \sqrt{2}}{2})}^{5}$

解题步骤 3.1.41.2

对 $i \sqrt{2}$ 运用乘积法则。

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{5 i \sqrt{2}}{4} + \frac{5 \sqrt{2}}{2} \cdot \frac{i^{4} {\sqrt{2}}^{4}}{2^{4}} + {(\frac{i \sqrt{2}}{2})}^{5}$

解题步骤 3.1.42

合并。

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{5 i \sqrt{2}}{4} + \frac{5 \sqrt{2} (i^{4} {\sqrt{2}}^{4})}{2 \cdot 2^{4}} + {(\frac{i \sqrt{2}}{2})}^{5}$

解题步骤 3.1.43

通过指数相加将 $\sqrt{2}$ 乘以 ${\sqrt{2}}^{4}$ 。

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解题步骤 3.1.43.1

移动 ${\sqrt{2}}^{4}$ 。

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{5 i \sqrt{2}}{4} + \frac{5 ({\sqrt{2}}^{4} \sqrt{2}) i^{4}}{2 \cdot 2^{4}} + {(\frac{i \sqrt{2}}{2})}^{5}$

解题步骤 3.1.43.2

将 ${\sqrt{2}}^{4}$ 乘以 $\sqrt{2}$ 。

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解题步骤 3.1.43.2.1

对 $\sqrt{2}$ 进行 $1$ 次方运算。

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{5 i \sqrt{2}}{4} + \frac{5 ({\sqrt{2}}^{4} {\sqrt{2}}^{1}) i^{4}}{2 \cdot 2^{4}} + {(\frac{i \sqrt{2}}{2})}^{5}$

解题步骤 3.1.43.2.2

使用幂法则 $a^{m} a^{n} = a^{m + n}$ 合并指数。

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{5 i \sqrt{2}}{4} + \frac{5 {\sqrt{2}}^{4 + 1} i^{4}}{2 \cdot 2^{4}} + {(\frac{i \sqrt{2}}{2})}^{5}$

解题步骤 3.1.43.3

将 $4$ 和 $1$ 相加。

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{5 i \sqrt{2}}{4} + \frac{5 {\sqrt{2}}^{5} i^{4}}{2 \cdot 2^{4}} + {(\frac{i \sqrt{2}}{2})}^{5}$

解题步骤 3.1.44

通过指数相加将 $2$ 乘以 $2^{4}$ 。

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解题步骤 3.1.44.1

将 $2$ 乘以 $2^{4}$ 。

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解题步骤 3.1.44.1.1

对 $2$ 进行 $1$ 次方运算。

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{5 i \sqrt{2}}{4} + \frac{5 {\sqrt{2}}^{5} i^{4}}{2^{1} \cdot 2^{4}} + {(\frac{i \sqrt{2}}{2})}^{5}$

解题步骤 3.1.44.1.2

使用幂法则 $a^{m} a^{n} = a^{m + n}$ 合并指数。

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{5 i \sqrt{2}}{4} + \frac{5 {\sqrt{2}}^{5} i^{4}}{2^{1 + 4}} + {(\frac{i \sqrt{2}}{2})}^{5}$

解题步骤 3.1.44.2

将 $1$ 和 $4$ 相加。

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{5 i \sqrt{2}}{4} + \frac{5 {\sqrt{2}}^{5} i^{4}}{2^{5}} + {(\frac{i \sqrt{2}}{2})}^{5}$

解题步骤 3.1.45

化简分子。

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解题步骤 3.1.45.1

将 ${\sqrt{2}}^{5}$ 重写为 $\sqrt{2^{5}}$ 。

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{5 i \sqrt{2}}{4} + \frac{5 \sqrt{2^{5}} i^{4}}{2^{5}} + {(\frac{i \sqrt{2}}{2})}^{5}$

解题步骤 3.1.45.2

对 $2$ 进行 $5$ 次方运算。

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{5 i \sqrt{2}}{4} + \frac{5 \sqrt{32} i^{4}}{2^{5}} + {(\frac{i \sqrt{2}}{2})}^{5}$

解题步骤 3.1.45.3

将 $32$ 重写为 $4^{2} \cdot 2$ 。

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解题步骤 3.1.45.3.1

从 $32$ 中分解出因数 $16$ 。

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{5 i \sqrt{2}}{4} + \frac{5 \sqrt{16 (2)} i^{4}}{2^{5}} + {(\frac{i \sqrt{2}}{2})}^{5}$

解题步骤 3.1.45.3.2

将 $16$ 重写为 $4^{2}$ 。

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{5 i \sqrt{2}}{4} + \frac{5 \sqrt{4^{2} \cdot 2} i^{4}}{2^{5}} + {(\frac{i \sqrt{2}}{2})}^{5}$

解题步骤 3.1.45.4

从根式下提出各项。

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{5 i \sqrt{2}}{4} + \frac{5 (4 \sqrt{2}) i^{4}}{2^{5}} + {(\frac{i \sqrt{2}}{2})}^{5}$

解题步骤 3.1.45.5

将 $i^{4}$ 重写为 $1$ 。

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解题步骤 3.1.45.5.1

将 $i^{4}$ 重写为 ${(i^{2})}^{2}$ 。

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{5 i \sqrt{2}}{4} + \frac{5 (4 \sqrt{2}) {(i^{2})}^{2}}{2^{5}} + {(\frac{i \sqrt{2}}{2})}^{5}$

解题步骤 3.1.45.5.2

将 $i^{2}$ 重写为 $- 1$ 。

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{5 i \sqrt{2}}{4} + \frac{5 (4 \sqrt{2}) {(- 1)}^{2}}{2^{5}} + {(\frac{i \sqrt{2}}{2})}^{5}$

解题步骤 3.1.45.5.3

对 $- 1$ 进行 $2$ 次方运算。

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{5 i \sqrt{2}}{4} + \frac{5 (4 \sqrt{2}) \cdot 1}{2^{5}} + {(\frac{i \sqrt{2}}{2})}^{5}$

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{5 i \sqrt{2}}{4} + \frac{5 \cdot 4 \sqrt{2} \cdot 1}{2^{5}} + {(\frac{i \sqrt{2}}{2})}^{5}$

解题步骤 3.1.45.6

合并指数。

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解题步骤 3.1.45.6.1

将 $5$ 乘以 $4$ 。

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{5 i \sqrt{2}}{4} + \frac{20 \sqrt{2} \cdot 1}{2^{5}} + {(\frac{i \sqrt{2}}{2})}^{5}$

解题步骤 3.1.45.6.2

将 $20$ 乘以 $1$ 。

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{5 i \sqrt{2}}{4} + \frac{20 \sqrt{2}}{2^{5}} + {(\frac{i \sqrt{2}}{2})}^{5}$

解题步骤 3.1.46

对 $2$ 进行 $5$ 次方运算。

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{5 i \sqrt{2}}{4} + \frac{20 \sqrt{2}}{32} + {(\frac{i \sqrt{2}}{2})}^{5}$

解题步骤 3.1.47

约去 $20$ 和 $32$ 的公因数。

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解题步骤 3.1.47.1

从 $20 \sqrt{2}$ 中分解出因数 $4$ 。

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{5 i \sqrt{2}}{4} + \frac{4 (5 \sqrt{2})}{32} + {(\frac{i \sqrt{2}}{2})}^{5}$

解题步骤 3.1.47.2

约去公因数。

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解题步骤 3.1.47.2.1

从 $32$ 中分解出因数 $4$ 。

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{5 i \sqrt{2}}{4} + \frac{4 (5 \sqrt{2})}{4 (8)} + {(\frac{i \sqrt{2}}{2})}^{5}$

解题步骤 3.1.47.2.2

约去公因数。

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{5 i \sqrt{2}}{4} + \frac{4 (5 \sqrt{2})}{4 \cdot 8} + {(\frac{i \sqrt{2}}{2})}^{5}$

解题步骤 3.1.47.2.3

重写表达式。

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{5 i \sqrt{2}}{4} + \frac{5 \sqrt{2}}{8} + {(\frac{i \sqrt{2}}{2})}^{5}$

解题步骤 3.1.48

使用幂法则 ${(a b)}^{n} = a^{n} b^{n}$ 分解指数。

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解题步骤 3.1.48.1

对 $\frac{i \sqrt{2}}{2}$ 运用乘积法则。

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{5 i \sqrt{2}}{4} + \frac{5 \sqrt{2}}{8} + \frac{{(i \sqrt{2})}^{5}}{2^{5}}$

解题步骤 3.1.48.2

对 $i \sqrt{2}$ 运用乘积法则。

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{5 i \sqrt{2}}{4} + \frac{5 \sqrt{2}}{8} + \frac{i^{5} {\sqrt{2}}^{5}}{2^{5}}$

解题步骤 3.1.49

化简分子。

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解题步骤 3.1.49.1

因式分解出 $i^{4}$ 。

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{5 i \sqrt{2}}{4} + \frac{5 \sqrt{2}}{8} + \frac{i^{4} i {\sqrt{2}}^{5}}{2^{5}}$

解题步骤 3.1.49.2

将 $i^{4}$ 重写为 $1$ 。

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解题步骤 3.1.49.2.1

将 $i^{4}$ 重写为 ${(i^{2})}^{2}$ 。

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{5 i \sqrt{2}}{4} + \frac{5 \sqrt{2}}{8} + \frac{{(i^{2})}^{2} i {\sqrt{2}}^{5}}{2^{5}}$

解题步骤 3.1.49.2.2

将 $i^{2}$ 重写为 $- 1$ 。

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{5 i \sqrt{2}}{4} + \frac{5 \sqrt{2}}{8} + \frac{{(- 1)}^{2} i {\sqrt{2}}^{5}}{2^{5}}$

解题步骤 3.1.49.2.3

对 $- 1$ 进行 $2$ 次方运算。

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{5 i \sqrt{2}}{4} + \frac{5 \sqrt{2}}{8} + \frac{1 i {\sqrt{2}}^{5}}{2^{5}}$

解题步骤 3.1.49.3

将 $i$ 乘以 $1$ 。

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{5 i \sqrt{2}}{4} + \frac{5 \sqrt{2}}{8} + \frac{i {\sqrt{2}}^{5}}{2^{5}}$

解题步骤 3.1.49.4

将 ${\sqrt{2}}^{5}$ 重写为 $\sqrt{2^{5}}$ 。

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{5 i \sqrt{2}}{4} + \frac{5 \sqrt{2}}{8} + \frac{i \sqrt{2^{5}}}{2^{5}}$

解题步骤 3.1.49.5

对 $2$ 进行 $5$ 次方运算。

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{5 i \sqrt{2}}{4} + \frac{5 \sqrt{2}}{8} + \frac{i \sqrt{32}}{2^{5}}$

解题步骤 3.1.49.6

将 $32$ 重写为 $4^{2} \cdot 2$ 。

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解题步骤 3.1.49.6.1

从 $32$ 中分解出因数 $16$ 。

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{5 i \sqrt{2}}{4} + \frac{5 \sqrt{2}}{8} + \frac{i \sqrt{16 (2)}}{2^{5}}$

解题步骤 3.1.49.6.2

将 $16$ 重写为 $4^{2}$ 。

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{5 i \sqrt{2}}{4} + \frac{5 \sqrt{2}}{8} + \frac{i \sqrt{4^{2} \cdot 2}}{2^{5}}$

解题步骤 3.1.49.7

从根式下提出各项。

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{5 i \sqrt{2}}{4} + \frac{5 \sqrt{2}}{8} + \frac{i \cdot 4 \sqrt{2}}{2^{5}}$

解题步骤 3.1.50

对 $2$ 进行 $5$ 次方运算。

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{5 i \sqrt{2}}{4} + \frac{5 \sqrt{2}}{8} + \frac{i \cdot 4 \sqrt{2}}{32}$

解题步骤 3.1.51

将 $4$ 移到 $i$ 的左侧。

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{5 i \sqrt{2}}{4} + \frac{5 \sqrt{2}}{8} + \frac{4 i \sqrt{2}}{32}$

解题步骤 3.1.52

约去 $4$ 和 $32$ 的公因数。

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解题步骤 3.1.52.1

从 $4 i \sqrt{2}$ 中分解出因数 $4$ 。

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{5 i \sqrt{2}}{4} + \frac{5 \sqrt{2}}{8} + \frac{4 (i \sqrt{2})}{32}$

解题步骤 3.1.52.2

约去公因数。

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解题步骤 3.1.52.2.1

从 $32$ 中分解出因数 $4$ 。

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{5 i \sqrt{2}}{4} + \frac{5 \sqrt{2}}{8} + \frac{4 (i \sqrt{2})}{4 \cdot 8}$

解题步骤 3.1.52.2.2

约去公因数。

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{5 i \sqrt{2}}{4} + \frac{5 \sqrt{2}}{8} + \frac{4 (i \sqrt{2})}{4 \cdot 8}$

解题步骤 3.1.52.2.3

重写表达式。

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{5 i \sqrt{2}}{4} + \frac{5 \sqrt{2}}{8} + \frac{i \sqrt{2}}{8}$

解题步骤 3.2

化简项。

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解题步骤 3.2.1

在公分母上合并分子。

$\frac{\sqrt{2} + 5 i \sqrt{2} + 5 \sqrt{2} + i \sqrt{2}}{8} + \frac{- 5 \sqrt{2}}{4} + \frac{- 5 i \sqrt{2}}{4}$

解题步骤 3.2.2

将 $\sqrt{2}$ 和 $5 \sqrt{2}$ 相加。

$\frac{5 i \sqrt{2} + 6 \sqrt{2} + i \sqrt{2}}{8} + \frac{- 5 \sqrt{2}}{4} + \frac{- 5 i \sqrt{2}}{4}$

解题步骤 3.2.3

将 $5 i \sqrt{2}$ 和 $i \sqrt{2}$ 相加。

$\frac{6 i \sqrt{2} + 6 \sqrt{2}}{8} + \frac{- 5 \sqrt{2}}{4} + \frac{- 5 i \sqrt{2}}{4}$

解题步骤 3.2.4

将 $6 i \sqrt{2}$ 和 $6 \sqrt{2}$ 重新排序。

$\frac{6 \sqrt{2} + 6 i \sqrt{2}}{8} + \frac{- 5 \sqrt{2}}{4} + \frac{- 5 i \sqrt{2}}{4}$

解题步骤 3.3

化简每一项。

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解题步骤 3.3.1

将负号移到分数的前面。

$\frac{6 \sqrt{2} + 6 i \sqrt{2}}{8} - \frac{(5) \sqrt{2}}{4} + \frac{- 5 i \sqrt{2}}{4}$

解题步骤 3.3.2

将负号移到分数的前面。

$\frac{6 \sqrt{2} + 6 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{5 i \sqrt{2}}{4}$

解题步骤 3.4

将 $\frac{6 \sqrt{2} + 6 i \sqrt{2}}{8}$ 和 $- \frac{5 \sqrt{2}}{4}$ 重新排序。

$- \frac{5 \sqrt{2}}{4} + \frac{6 \sqrt{2} + 6 i \sqrt{2}}{8} - \frac{5 i \sqrt{2}}{4}$

解题步骤 4

这是复数的三角函数形式，其中 $| z |$ 是模数， $θ$ 是复平面上形成的夹角。

$z = a + b i = | z | (\cos (θ) + i \sin (θ))$

解题步骤 5

复数的模是复平面上距离原点的距离。

当 $z = a + b i$ 时， $| z | = \sqrt{a^{2} + b^{2}}$

解题步骤 6

代入 $a = - \frac{5 \sqrt{2}}{4}$ 和 $b = \frac{1}{8}$ 的实际值。

$| z | = \sqrt{{(\frac{1}{8})}^{2} + {(- \frac{5 \sqrt{2}}{4})}^{2}}$

解题步骤 7

求 $| z |$ 。

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解题步骤 7.1

对 $\frac{1}{8}$ 运用乘积法则。

$| z | = \sqrt{\frac{1^{2}}{8^{2}} + {(- \frac{5 \sqrt{2}}{4})}^{2}}$

解题步骤 7.2

一的任意次幂都为一。

$| z | = \sqrt{\frac{1}{8^{2}} + {(- \frac{5 \sqrt{2}}{4})}^{2}}$

解题步骤 7.3

对 $8$ 进行 $2$ 次方运算。

$| z | = \sqrt{\frac{1}{64} + {(- \frac{5 \sqrt{2}}{4})}^{2}}$

解题步骤 7.4

使用幂法则 ${(a b)}^{n} = a^{n} b^{n}$ 分解指数。

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解题步骤 7.4.1

对 $- \frac{5 \sqrt{2}}{4}$ 运用乘积法则。

$| z | = \sqrt{\frac{1}{64} + {(- 1)}^{2} {(\frac{5 \sqrt{2}}{4})}^{2}}$

解题步骤 7.4.2

对 $\frac{5 \sqrt{2}}{4}$ 运用乘积法则。

$| z | = \sqrt{\frac{1}{64} + {(- 1)}^{2} (\frac{{(5 \sqrt{2})}^{2}}{4^{2}})}$

解题步骤 7.4.3

对 $5 \sqrt{2}$ 运用乘积法则。

$| z | = \sqrt{\frac{1}{64} + {(- 1)}^{2} (\frac{5^{2} {\sqrt{2}}^{2}}{4^{2}})}$

解题步骤 7.5

化简表达式。

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解题步骤 7.5.1

对 $- 1$ 进行 $2$ 次方运算。

$| z | = \sqrt{\frac{1}{64} + 1 (\frac{5^{2} {\sqrt{2}}^{2}}{4^{2}})}$

解题步骤 7.5.2

将 $\frac{5^{2} {\sqrt{2}}^{2}}{4^{2}}$ 乘以 $1$ 。

$| z | = \sqrt{\frac{1}{64} + \frac{5^{2} {\sqrt{2}}^{2}}{4^{2}}}$

解题步骤 7.6

化简分子。

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解题步骤 7.6.1

对 $5$ 进行 $2$ 次方运算。

$| z | = \sqrt{\frac{1}{64} + \frac{25 {\sqrt{2}}^{2}}{4^{2}}}$

解题步骤 7.6.2

将 ${\sqrt{2}}^{2}$ 重写为 $2$ 。

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解题步骤 7.6.2.1

使用 $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ ，将 $\sqrt{2}$ 重写成 $2^{\frac{1}{2}}$ 。

$| z | = \sqrt{\frac{1}{64} + \frac{25 {(2^{\frac{1}{2}})}^{2}}{4^{2}}}$

解题步骤 7.6.2.2

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

$| z | = \sqrt{\frac{1}{64} + \frac{25 \cdot 2^{\frac{1}{2} \cdot 2}}{4^{2}}}$

解题步骤 7.6.2.3

组合 $\frac{1}{2}$ 和 $2$ 。

$| z | = \sqrt{\frac{1}{64} + \frac{25 \cdot 2^{\frac{2}{2}}}{4^{2}}}$

解题步骤 7.6.2.4

约去 $2$ 的公因数。

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解题步骤 7.6.2.4.1

约去公因数。

$| z | = \sqrt{\frac{1}{64} + \frac{25 \cdot 2^{\frac{2}{2}}}{4^{2}}}$

解题步骤 7.6.2.4.2

重写表达式。

$| z | = \sqrt{\frac{1}{64} + \frac{25 \cdot 2}{4^{2}}}$

解题步骤 7.6.2.5

计算指数。

$| z | = \sqrt{\frac{1}{64} + \frac{25 \cdot 2}{4^{2}}}$

解题步骤 7.7

通过约去公因数来化简表达式。

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解题步骤 7.7.1

对 $4$ 进行 $2$ 次方运算。

$| z | = \sqrt{\frac{1}{64} + \frac{25 \cdot 2}{16}}$

解题步骤 7.7.2

将 $25$ 乘以 $2$ 。

$| z | = \sqrt{\frac{1}{64} + \frac{50}{16}}$

解题步骤 7.7.3

约去 $50$ 和 $16$ 的公因数。

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解题步骤 7.7.3.1

从 $50$ 中分解出因数 $2$ 。

$| z | = \sqrt{\frac{1}{64} + \frac{2 (25)}{16}}$

解题步骤 7.7.3.2

约去公因数。

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解题步骤 7.7.3.2.1

从 $16$ 中分解出因数 $2$ 。

$| z | = \sqrt{\frac{1}{64} + \frac{2 \cdot 25}{2 \cdot 8}}$

解题步骤 7.7.3.2.2

约去公因数。

$| z | = \sqrt{\frac{1}{64} + \frac{2 \cdot 25}{2 \cdot 8}}$

解题步骤 7.7.3.2.3

重写表达式。

$| z | = \sqrt{\frac{1}{64} + \frac{25}{8}}$

解题步骤 7.8

要将 $\frac{25}{8}$ 写成带有公分母的分数，请乘以 $\frac{8}{8}$ 。

$| z | = \sqrt{\frac{1}{64} + \frac{25}{8} \cdot \frac{8}{8}}$

解题步骤 7.9

通过与 $1$ 的合适因数相乘，将每一个表达式写成具有公分母 $64$ 的形式。

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解题步骤 7.9.1

将 $\frac{25}{8}$ 乘以 $\frac{8}{8}$ 。

$| z | = \sqrt{\frac{1}{64} + \frac{25 \cdot 8}{8 \cdot 8}}$

解题步骤 7.9.2

将 $8$ 乘以 $8$ 。

$| z | = \sqrt{\frac{1}{64} + \frac{25 \cdot 8}{64}}$

解题步骤 7.10

在公分母上合并分子。

$| z | = \sqrt{\frac{1 + 25 \cdot 8}{64}}$

解题步骤 7.11

化简分子。

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解题步骤 7.11.1

将 $25$ 乘以 $8$ 。

$| z | = \sqrt{\frac{1 + 200}{64}}$

解题步骤 7.11.2

将 $1$ 和 $200$ 相加。

$| z | = \sqrt{\frac{201}{64}}$

解题步骤 7.12

将 $\sqrt{\frac{201}{64}}$ 重写为 $\frac{\sqrt{201}}{\sqrt{64}}$ 。

$| z | = \frac{\sqrt{201}}{\sqrt{64}}$

解题步骤 7.13

化简分母。

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解题步骤 7.13.1

将 $64$ 重写为 $8^{2}$ 。

$| z | = \frac{\sqrt{201}}{\sqrt{8^{2}}}$

解题步骤 7.13.2

假设各项均为正实数，从根式下提出各项。

$| z | = \frac{\sqrt{201}}{8}$

解题步骤 8

复平面上点的角为复数部分除以实数部分的逆正切。

$θ = \arctan (\frac{\frac{1}{8}}{- \frac{5 \sqrt{2}}{4}})$

解题步骤 9

因为 $\frac{\frac{1}{8}}{- \frac{5 \sqrt{2}}{4}}$ 的反正切得出位于第二象限的一个角，所以其角度为 $3.07099947$ 。

$θ = 3.07099947$

解题步骤 10

代入 $θ = 3.07099947$ 和 $| z | = \frac{\sqrt{201}}{8}$ 的值。

$\frac{\sqrt{201}}{8 (\cos (3.07099947) + i \sin (3.07099947))}$

微积分学 示例

微积分学示例