Preálgebra Ejemplos

$\frac{2 \sqrt{7}}{3 \sqrt{5 - \sqrt{2}}} \cdot \frac{\sqrt{3}}{\sqrt{5 - 3 \sqrt{2}}}$

Paso 1

Multiplica $\frac{2 \sqrt{7}}{3 \sqrt{5 - \sqrt{2}}}$ por $\frac{\sqrt{5 - \sqrt{2}}}{\sqrt{5 - \sqrt{2}}}$ .

$\frac{2 \sqrt{7}}{3 \sqrt{5 - \sqrt{2}}} \cdot \frac{\sqrt{5 - \sqrt{2}}}{\sqrt{5 - \sqrt{2}}} \cdot \frac{\sqrt{3}}{\sqrt{5 - 3 \sqrt{2}}}$

Paso 2

Simplifica los términos.

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Paso 2.1

Combina y simplifica el denominador.

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Paso 2.1.1

Multiplica $\frac{2 \sqrt{7}}{3 \sqrt{5 - \sqrt{2}}}$ por $\frac{\sqrt{5 - \sqrt{2}}}{\sqrt{5 - \sqrt{2}}}$ .

$\frac{2 \sqrt{7} \sqrt{5 - \sqrt{2}}}{3 \sqrt{5 - \sqrt{2}} \sqrt{5 - \sqrt{2}}} \cdot \frac{\sqrt{3}}{\sqrt{5 - 3 \sqrt{2}}}$

Paso 2.1.2

Mueve $\sqrt{5 - \sqrt{2}}$ .

$\frac{2 \sqrt{7} \sqrt{5 - \sqrt{2}}}{3 (\sqrt{5 - \sqrt{2}} \sqrt{5 - \sqrt{2}})} \cdot \frac{\sqrt{3}}{\sqrt{5 - 3 \sqrt{2}}}$

Paso 2.1.3

Eleva $\sqrt{5 - \sqrt{2}}$ a la potencia de $1$ .

$\frac{2 \sqrt{7} \sqrt{5 - \sqrt{2}}}{3 ({\sqrt{5 - \sqrt{2}}}^{1} \sqrt{5 - \sqrt{2}})} \cdot \frac{\sqrt{3}}{\sqrt{5 - 3 \sqrt{2}}}$

Paso 2.1.4

Eleva $\sqrt{5 - \sqrt{2}}$ a la potencia de $1$ .

$\frac{2 \sqrt{7} \sqrt{5 - \sqrt{2}}}{3 ({\sqrt{5 - \sqrt{2}}}^{1} {\sqrt{5 - \sqrt{2}}}^{1})} \cdot \frac{\sqrt{3}}{\sqrt{5 - 3 \sqrt{2}}}$

Paso 2.1.5

Usa la regla de la potencia $a^{m} a^{n} = a^{m + n}$ para combinar exponentes.

$\frac{2 \sqrt{7} \sqrt{5 - \sqrt{2}}}{3 {\sqrt{5 - \sqrt{2}}}^{1 + 1}} \cdot \frac{\sqrt{3}}{\sqrt{5 - 3 \sqrt{2}}}$

Paso 2.1.6

Suma $1$ y $1$ .

$\frac{2 \sqrt{7} \sqrt{5 - \sqrt{2}}}{3 {\sqrt{5 - \sqrt{2}}}^{2}} \cdot \frac{\sqrt{3}}{\sqrt{5 - 3 \sqrt{2}}}$

Paso 2.1.7

Reescribe ${\sqrt{5 - \sqrt{2}}}^{2}$ como $5 - \sqrt{2}$ .

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Paso 2.1.7.1

Usa $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ para reescribir $\sqrt{5 - \sqrt{2}}$ como ${(5 - \sqrt{2})}^{\frac{1}{2}}$ .

$\frac{2 \sqrt{7} \sqrt{5 - \sqrt{2}}}{3 {({(5 - \sqrt{2})}^{\frac{1}{2}})}^{2}} \cdot \frac{\sqrt{3}}{\sqrt{5 - 3 \sqrt{2}}}$

Paso 2.1.7.2

Aplica la regla de la potencia y multiplica los exponentes, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{2 \sqrt{7} \sqrt{5 - \sqrt{2}}}{3 {(5 - \sqrt{2})}^{\frac{1}{2} \cdot 2}} \cdot \frac{\sqrt{3}}{\sqrt{5 - 3 \sqrt{2}}}$

Paso 2.1.7.3

Combina $\frac{1}{2}$ y $2$ .

$\frac{2 \sqrt{7} \sqrt{5 - \sqrt{2}}}{3 {(5 - \sqrt{2})}^{\frac{2}{2}}} \cdot \frac{\sqrt{3}}{\sqrt{5 - 3 \sqrt{2}}}$

Paso 2.1.7.4

Cancela el factor común de $2$ .

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Paso 2.1.7.4.1

Cancela el factor común.

$\frac{2 \sqrt{7} \sqrt{5 - \sqrt{2}}}{3 {(5 - \sqrt{2})}^{\frac{2}{2}}} \cdot \frac{\sqrt{3}}{\sqrt{5 - 3 \sqrt{2}}}$

Paso 2.1.7.4.2

Reescribe la expresión.

$\frac{2 \sqrt{7} \sqrt{5 - \sqrt{2}}}{3 {(5 - \sqrt{2})}^{1}} \cdot \frac{\sqrt{3}}{\sqrt{5 - 3 \sqrt{2}}}$

Paso 2.1.7.5

Simplifica.

$\frac{2 \sqrt{7} \sqrt{5 - \sqrt{2}}}{3 (5 - \sqrt{2})} \cdot \frac{\sqrt{3}}{\sqrt{5 - 3 \sqrt{2}}}$

Paso 2.2

Combina con la regla del producto para radicales.

$\frac{2 \sqrt{(5 - \sqrt{2}) \cdot 7}}{3 (5 - \sqrt{2})} \cdot \frac{\sqrt{3}}{\sqrt{5 - 3 \sqrt{2}}}$

Paso 3

Simplifica el numerador.

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Paso 3.1

Aplica la propiedad distributiva.

$\frac{2 \sqrt{5 \cdot 7 - \sqrt{2} \cdot 7}}{3 (5 - \sqrt{2})} \cdot \frac{\sqrt{3}}{\sqrt{5 - 3 \sqrt{2}}}$

Paso 3.2

Multiplica $5$ por $7$ .

$\frac{2 \sqrt{35 - \sqrt{2} \cdot 7}}{3 (5 - \sqrt{2})} \cdot \frac{\sqrt{3}}{\sqrt{5 - 3 \sqrt{2}}}$

Paso 3.3

Multiplica $7$ por $- 1$ .

$\frac{2 \sqrt{35 - 7 \sqrt{2}}}{3 (5 - \sqrt{2})} \cdot \frac{\sqrt{3}}{\sqrt{5 - 3 \sqrt{2}}}$

Paso 4

Multiplica $\frac{2 \sqrt{35 - 7 \sqrt{2}}}{3 (5 - \sqrt{2})}$ por $\frac{5 + \sqrt{2}}{5 + \sqrt{2}}$ .

$\frac{2 \sqrt{35 - 7 \sqrt{2}}}{3 (5 - \sqrt{2})} \cdot \frac{5 + \sqrt{2}}{5 + \sqrt{2}} \cdot \frac{\sqrt{3}}{\sqrt{5 - 3 \sqrt{2}}}$

Paso 5

Simplifica los términos.

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Paso 5.1

Multiplica $\frac{2 \sqrt{35 - 7 \sqrt{2}}}{3 (5 - \sqrt{2})}$ por $\frac{5 + \sqrt{2}}{5 + \sqrt{2}}$ .

$\frac{2 \sqrt{35 - 7 \sqrt{2}} (5 + \sqrt{2})}{3 (5 - \sqrt{2}) (5 + \sqrt{2})} \cdot \frac{\sqrt{3}}{\sqrt{5 - 3 \sqrt{2}}}$

Paso 5.2

Simplifica la expresión.

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Paso 5.2.1

Mueve $5 - \sqrt{2}$ .

$\frac{2 \sqrt{35 - 7 \sqrt{2}} (5 + \sqrt{2})}{3 ((5 - \sqrt{2}) (5 + \sqrt{2}))} \cdot \frac{\sqrt{3}}{\sqrt{5 - 3 \sqrt{2}}}$

Paso 5.2.2

Expande el denominador con el método PEIU.

$\frac{2 \sqrt{35 - 7 \sqrt{2}} (5 + \sqrt{2})}{3 (25 + 5 \sqrt{2} - 5 \sqrt{2} - {\sqrt{2}}^{2})} \cdot \frac{\sqrt{3}}{\sqrt{5 - 3 \sqrt{2}}}$

Paso 5.2.3

Simplifica.

$\frac{2 \sqrt{35 - 7 \sqrt{2}} (5 + \sqrt{2})}{3 \cdot 23} \cdot \frac{\sqrt{3}}{\sqrt{5 - 3 \sqrt{2}}}$

Paso 5.2.4

Multiplica $3$ por $23$ .

$\frac{2 \sqrt{35 - 7 \sqrt{2}} (5 + \sqrt{2})}{69} \cdot \frac{\sqrt{3}}{\sqrt{5 - 3 \sqrt{2}}}$

Paso 5.2.5

Agrupa $5 + \sqrt{2}$ y $\sqrt{35 - 7 \sqrt{2}}$ .

$\frac{2 ((5 + \sqrt{2}) \sqrt{35 - 7 \sqrt{2}})}{69} \cdot \frac{\sqrt{3}}{\sqrt{5 - 3 \sqrt{2}}}$

Paso 5.3

Aplica la propiedad distributiva.

$\frac{2 (5 \sqrt{35 - 7 \sqrt{2}} + \sqrt{2} \sqrt{35 - 7 \sqrt{2}})}{69} \cdot \frac{\sqrt{3}}{\sqrt{5 - 3 \sqrt{2}}}$

Paso 5.4

Combina con la regla del producto para radicales.

$\frac{2 (5 \sqrt{35 - 7 \sqrt{2}} + \sqrt{2 (35 - 7 \sqrt{2})})}{69} \cdot \frac{\sqrt{3}}{\sqrt{5 - 3 \sqrt{2}}}$

Paso 5.5

Multiplica $\frac{2 (5 \sqrt{35 - 7 \sqrt{2}} + \sqrt{2 (35 - 7 \sqrt{2})})}{69}$ por $\frac{\sqrt{3}}{\sqrt{5 - 3 \sqrt{2}}}$ .

$\frac{2 (5 \sqrt{35 - 7 \sqrt{2}} + \sqrt{2 (35 - 7 \sqrt{2})}) \sqrt{3}}{69 \sqrt{5 - 3 \sqrt{2}}}$

Paso 5.6

Agrupa $\sqrt{3}$ y $5 \sqrt{35 - 7 \sqrt{2}} + \sqrt{2 (35 - 7 \sqrt{2})}$ .

$\frac{2 (\sqrt{3} (5 \sqrt{35 - 7 \sqrt{2}} + \sqrt{2 (35 - 7 \sqrt{2})}))}{69 \sqrt{5 - 3 \sqrt{2}}}$

Paso 5.7

Aplica la propiedad distributiva.

$\frac{2 (\sqrt{3} (5 \sqrt{35 - 7 \sqrt{2}}) + \sqrt{3} \sqrt{2 (35 - 7 \sqrt{2})})}{69 \sqrt{5 - 3 \sqrt{2}}}$

Paso 5.8

Combina con la regla del producto para radicales.

$\frac{2 (5 \sqrt{3 (35 - 7 \sqrt{2})} + \sqrt{3} \sqrt{2 (35 - 7 \sqrt{2})})}{69 \sqrt{5 - 3 \sqrt{2}}}$

Paso 5.9

Combina con la regla del producto para radicales.

$\frac{2 (5 \sqrt{3 (35 - 7 \sqrt{2})} + \sqrt{3 (2 (35 - 7 \sqrt{2}))})}{69 \sqrt{5 - 3 \sqrt{2}}}$

Paso 6

Multiplica $3$ por $2$ .

$\frac{2 (5 \sqrt{3 (35 - 7 \sqrt{2})} + \sqrt{6 (35 - 7 \sqrt{2})})}{69 \sqrt{5 - 3 \sqrt{2}}}$

Paso 7

Multiplica $\frac{2 (5 \sqrt{3 (35 - 7 \sqrt{2})} + \sqrt{6 (35 - 7 \sqrt{2})})}{69 \sqrt{5 - 3 \sqrt{2}}}$ por $\frac{\sqrt{5 - 3 \sqrt{2}}}{\sqrt{5 - 3 \sqrt{2}}}$ .

$\frac{2 (5 \sqrt{3 (35 - 7 \sqrt{2})} + \sqrt{6 (35 - 7 \sqrt{2})})}{69 \sqrt{5 - 3 \sqrt{2}}} \cdot \frac{\sqrt{5 - 3 \sqrt{2}}}{\sqrt{5 - 3 \sqrt{2}}}$

Paso 8

Combina y simplifica el denominador.

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Paso 8.1

Multiplica $\frac{2 (5 \sqrt{3 (35 - 7 \sqrt{2})} + \sqrt{6 (35 - 7 \sqrt{2})})}{69 \sqrt{5 - 3 \sqrt{2}}}$ por $\frac{\sqrt{5 - 3 \sqrt{2}}}{\sqrt{5 - 3 \sqrt{2}}}$ .

$\frac{2 (5 \sqrt{3 (35 - 7 \sqrt{2})} + \sqrt{6 (35 - 7 \sqrt{2})}) \sqrt{5 - 3 \sqrt{2}}}{69 \sqrt{5 - 3 \sqrt{2}} \sqrt{5 - 3 \sqrt{2}}}$

Paso 8.2

Mueve $\sqrt{5 - 3 \sqrt{2}}$ .

$\frac{2 (5 \sqrt{3 (35 - 7 \sqrt{2})} + \sqrt{6 (35 - 7 \sqrt{2})}) \sqrt{5 - 3 \sqrt{2}}}{69 (\sqrt{5 - 3 \sqrt{2}} \sqrt{5 - 3 \sqrt{2}})}$

Paso 8.3

Eleva $\sqrt{5 - 3 \sqrt{2}}$ a la potencia de $1$ .

$\frac{2 (5 \sqrt{3 (35 - 7 \sqrt{2})} + \sqrt{6 (35 - 7 \sqrt{2})}) \sqrt{5 - 3 \sqrt{2}}}{69 ({\sqrt{5 - 3 \sqrt{2}}}^{1} \sqrt{5 - 3 \sqrt{2}})}$

Paso 8.4

Eleva $\sqrt{5 - 3 \sqrt{2}}$ a la potencia de $1$ .

$\frac{2 (5 \sqrt{3 (35 - 7 \sqrt{2})} + \sqrt{6 (35 - 7 \sqrt{2})}) \sqrt{5 - 3 \sqrt{2}}}{69 ({\sqrt{5 - 3 \sqrt{2}}}^{1} {\sqrt{5 - 3 \sqrt{2}}}^{1})}$

Paso 8.5

Usa la regla de la potencia $a^{m} a^{n} = a^{m + n}$ para combinar exponentes.

$\frac{2 (5 \sqrt{3 (35 - 7 \sqrt{2})} + \sqrt{6 (35 - 7 \sqrt{2})}) \sqrt{5 - 3 \sqrt{2}}}{69 {\sqrt{5 - 3 \sqrt{2}}}^{1 + 1}}$

Paso 8.6

Suma $1$ y $1$ .

$\frac{2 (5 \sqrt{3 (35 - 7 \sqrt{2})} + \sqrt{6 (35 - 7 \sqrt{2})}) \sqrt{5 - 3 \sqrt{2}}}{69 {\sqrt{5 - 3 \sqrt{2}}}^{2}}$

Paso 8.7

Reescribe ${\sqrt{5 - 3 \sqrt{2}}}^{2}$ como $5 - 3 \sqrt{2}$ .

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Paso 8.7.1

Usa $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ para reescribir $\sqrt{5 - 3 \sqrt{2}}$ como ${(5 - 3 \sqrt{2})}^{\frac{1}{2}}$ .

$\frac{2 (5 \sqrt{3 (35 - 7 \sqrt{2})} + \sqrt{6 (35 - 7 \sqrt{2})}) \sqrt{5 - 3 \sqrt{2}}}{69 {({(5 - 3 \sqrt{2})}^{\frac{1}{2}})}^{2}}$

Paso 8.7.2

Aplica la regla de la potencia y multiplica los exponentes, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{2 (5 \sqrt{3 (35 - 7 \sqrt{2})} + \sqrt{6 (35 - 7 \sqrt{2})}) \sqrt{5 - 3 \sqrt{2}}}{69 {(5 - 3 \sqrt{2})}^{\frac{1}{2} \cdot 2}}$

Paso 8.7.3

Combina $\frac{1}{2}$ y $2$ .

$\frac{2 (5 \sqrt{3 (35 - 7 \sqrt{2})} + \sqrt{6 (35 - 7 \sqrt{2})}) \sqrt{5 - 3 \sqrt{2}}}{69 {(5 - 3 \sqrt{2})}^{\frac{2}{2}}}$

Paso 8.7.4

Cancela el factor común de $2$ .

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Paso 8.7.4.1

Cancela el factor común.

$\frac{2 (5 \sqrt{3 (35 - 7 \sqrt{2})} + \sqrt{6 (35 - 7 \sqrt{2})}) \sqrt{5 - 3 \sqrt{2}}}{69 {(5 - 3 \sqrt{2})}^{\frac{2}{2}}}$

Paso 8.7.4.2

Reescribe la expresión.

$\frac{2 (5 \sqrt{3 (35 - 7 \sqrt{2})} + \sqrt{6 (35 - 7 \sqrt{2})}) \sqrt{5 - 3 \sqrt{2}}}{69 {(5 - 3 \sqrt{2})}^{1}}$

Paso 8.7.5

Simplifica.

$\frac{2 (5 \sqrt{3 (35 - 7 \sqrt{2})} + \sqrt{6 (35 - 7 \sqrt{2})}) \sqrt{5 - 3 \sqrt{2}}}{69 (5 - 3 \sqrt{2})}$

Paso 9

Agrupa $\sqrt{5 - 3 \sqrt{2}}$ y $5 \sqrt{3 (35 - 7 \sqrt{2})} + \sqrt{6 (35 - 7 \sqrt{2})}$ .

$\frac{2 (\sqrt{5 - 3 \sqrt{2}} (5 \sqrt{3 (35 - 7 \sqrt{2})} + \sqrt{6 (35 - 7 \sqrt{2})}))}{69 (5 - 3 \sqrt{2})}$

Paso 10

Aplica la propiedad distributiva.

$\frac{2 (\sqrt{5 - 3 \sqrt{2}} (5 \sqrt{3 (35 - 7 \sqrt{2})}) + \sqrt{5 - 3 \sqrt{2}} \sqrt{6 (35 - 7 \sqrt{2})})}{69 (5 - 3 \sqrt{2})}$

Paso 11

Combina con la regla del producto para radicales.

$\frac{2 (5 \sqrt{(5 - 3 \sqrt{2}) \cdot 3 (35 - 7 \sqrt{2})} + \sqrt{5 - 3 \sqrt{2}} \sqrt{6 (35 - 7 \sqrt{2})})}{69 (5 - 3 \sqrt{2})}$

Paso 12

Combina con la regla del producto para radicales.

$\frac{2 (5 \sqrt{(5 - 3 \sqrt{2}) \cdot 3 (35 - 7 \sqrt{2})} + \sqrt{(5 - 3 \sqrt{2}) \cdot 6 (35 - 7 \sqrt{2})})}{69 (5 - 3 \sqrt{2})}$

Paso 13

Reordena.

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Paso 13.1

Mueve $3$ a la izquierda de $5 - 3 \sqrt{2}$ .

$\frac{2 (5 \sqrt{3 \cdot (5 - 3 \sqrt{2}) (35 - 7 \sqrt{2})} + \sqrt{(5 - 3 \sqrt{2}) \cdot 6 (35 - 7 \sqrt{2})})}{69 (5 - 3 \sqrt{2})}$

Paso 13.2

Mueve $6$ a la izquierda de $5 - 3 \sqrt{2}$ .

$\frac{2 (5 \sqrt{3 \cdot (5 - 3 \sqrt{2}) (35 - 7 \sqrt{2})} + \sqrt{6 \cdot (5 - 3 \sqrt{2}) (35 - 7 \sqrt{2})})}{69 (5 - 3 \sqrt{2})}$

Paso 14

Multiplica $\frac{2 (5 \sqrt{3 \cdot (5 - 3 \sqrt{2}) (35 - 7 \sqrt{2})} + \sqrt{6 \cdot (5 - 3 \sqrt{2}) (35 - 7 \sqrt{2})})}{69 (5 - 3 \sqrt{2})}$ por $\frac{5 + 3 \sqrt{2}}{5 + 3 \sqrt{2}}$ .

$\frac{2 (5 \sqrt{3 \cdot (5 - 3 \sqrt{2}) (35 - 7 \sqrt{2})} + \sqrt{6 \cdot (5 - 3 \sqrt{2}) (35 - 7 \sqrt{2})})}{69 (5 - 3 \sqrt{2})} \cdot \frac{5 + 3 \sqrt{2}}{5 + 3 \sqrt{2}}$

Paso 15

Combina fracciones.

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Paso 15.1

Multiplica $\frac{2 (5 \sqrt{3 \cdot (5 - 3 \sqrt{2}) (35 - 7 \sqrt{2})} + \sqrt{6 \cdot (5 - 3 \sqrt{2}) (35 - 7 \sqrt{2})})}{69 (5 - 3 \sqrt{2})}$ por $\frac{5 + 3 \sqrt{2}}{5 + 3 \sqrt{2}}$ .

$\frac{2 (5 \sqrt{3 \cdot (5 - 3 \sqrt{2}) (35 - 7 \sqrt{2})} + \sqrt{6 \cdot (5 - 3 \sqrt{2}) (35 - 7 \sqrt{2})}) (5 + 3 \sqrt{2})}{69 (5 - 3 \sqrt{2}) (5 + 3 \sqrt{2})}$

Paso 15.2

Simplifica la expresión.

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Paso 15.2.1

Mueve $5 - 3 \sqrt{2}$ .

$\frac{2 (5 \sqrt{3 \cdot (5 - 3 \sqrt{2}) (35 - 7 \sqrt{2})} + \sqrt{6 \cdot (5 - 3 \sqrt{2}) (35 - 7 \sqrt{2})}) (5 + 3 \sqrt{2})}{69 ((5 - 3 \sqrt{2}) (5 + 3 \sqrt{2}))}$

Paso 15.2.2

Expande el denominador con el método PEIU.

$\frac{2 (5 \sqrt{3 \cdot (5 - 3 \sqrt{2}) (35 - 7 \sqrt{2})} + \sqrt{6 \cdot (5 - 3 \sqrt{2}) (35 - 7 \sqrt{2})}) (5 + 3 \sqrt{2})}{69 (25 + 15 \sqrt{2} - 15 \sqrt{2} - 9 {\sqrt{2}}^{2})}$

Paso 15.2.3

Simplifica.

$\frac{2 (5 \sqrt{3 \cdot (5 - 3 \sqrt{2}) (35 - 7 \sqrt{2})} + \sqrt{6 \cdot (5 - 3 \sqrt{2}) (35 - 7 \sqrt{2})}) (5 + 3 \sqrt{2})}{69 \cdot 7}$

Paso 15.2.4

Multiplica $69$ por $7$ .

$\frac{2 (5 \sqrt{3 \cdot (5 - 3 \sqrt{2}) (35 - 7 \sqrt{2})} + \sqrt{6 \cdot (5 - 3 \sqrt{2}) (35 - 7 \sqrt{2})}) (5 + 3 \sqrt{2})}{483}$

Paso 15.2.5

Agrupa $5 + 3 \sqrt{2}$ y $5 \sqrt{3 \cdot (5 - 3 \sqrt{2}) (35 - 7 \sqrt{2})} + \sqrt{6 \cdot (5 - 3 \sqrt{2}) (35 - 7 \sqrt{2})}$ .

$\frac{2 ((5 + 3 \sqrt{2}) (5 \sqrt{3 \cdot (5 - 3 \sqrt{2}) (35 - 7 \sqrt{2})} + \sqrt{6 \cdot (5 - 3 \sqrt{2}) (35 - 7 \sqrt{2})}))}{483}$

Paso 16

Expande $(5 + 3 \sqrt{2}) (5 \sqrt{3 \cdot (5 - 3 \sqrt{2}) (35 - 7 \sqrt{2})} + \sqrt{6 \cdot (5 - 3 \sqrt{2}) (35 - 7 \sqrt{2})})$ con el método PEIU (primero, exterior, interior, ultimo).

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Paso 16.1

Aplica la propiedad distributiva.

$\frac{2 (5 (5 \sqrt{3 \cdot (5 - 3 \sqrt{2}) (35 - 7 \sqrt{2})} + \sqrt{6 \cdot (5 - 3 \sqrt{2}) (35 - 7 \sqrt{2})}) + 3 \sqrt{2} (5 \sqrt{3 \cdot (5 - 3 \sqrt{2}) (35 - 7 \sqrt{2})} + \sqrt{6 \cdot (5 - 3 \sqrt{2}) (35 - 7 \sqrt{2})}))}{483}$

Paso 16.2

Aplica la propiedad distributiva.

$\frac{2 (5 (5 \sqrt{3 \cdot (5 - 3 \sqrt{2}) (35 - 7 \sqrt{2})}) + 5 \sqrt{6 \cdot (5 - 3 \sqrt{2}) (35 - 7 \sqrt{2})} + 3 \sqrt{2} (5 \sqrt{3 \cdot (5 - 3 \sqrt{2}) (35 - 7 \sqrt{2})} + \sqrt{6 \cdot (5 - 3 \sqrt{2}) (35 - 7 \sqrt{2})}))}{483}$

Paso 16.3

Aplica la propiedad distributiva.

$\frac{2 (5 (5 \sqrt{3 \cdot (5 - 3 \sqrt{2}) (35 - 7 \sqrt{2})}) + 5 \sqrt{6 \cdot (5 - 3 \sqrt{2}) (35 - 7 \sqrt{2})} + 3 \sqrt{2} (5 \sqrt{3 \cdot (5 - 3 \sqrt{2}) (35 - 7 \sqrt{2})}) + 3 \sqrt{2} \sqrt{6 \cdot (5 - 3 \sqrt{2}) (35 - 7 \sqrt{2})})}{483}$

Paso 17

Simplifica y combina los términos similares.

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Paso 17.1

Simplifica cada término.

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Paso 17.1.1

Multiplica $5$ por $5$ .

$\frac{2 (25 \sqrt{3 \cdot (5 - 3 \sqrt{2}) (35 - 7 \sqrt{2})} + 5 \sqrt{6 \cdot (5 - 3 \sqrt{2}) (35 - 7 \sqrt{2})} + 3 \sqrt{2} (5 \sqrt{3 \cdot (5 - 3 \sqrt{2}) (35 - 7 \sqrt{2})}) + 3 \sqrt{2} \sqrt{6 \cdot (5 - 3 \sqrt{2}) (35 - 7 \sqrt{2})})}{483}$

Paso 17.1.2

Multiplica $3 \sqrt{2} (5 \sqrt{3 \cdot (5 - 3 \sqrt{2}) (35 - 7 \sqrt{2})})$ .

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Paso 17.1.2.1

Multiplica $5$ por $3$ .

$\frac{2 (25 \sqrt{3 \cdot (5 - 3 \sqrt{2}) (35 - 7 \sqrt{2})} + 5 \sqrt{6 \cdot (5 - 3 \sqrt{2}) (35 - 7 \sqrt{2})} + 15 \sqrt{2} \sqrt{3 \cdot (5 - 3 \sqrt{2}) (35 - 7 \sqrt{2})} + 3 \sqrt{2} \sqrt{6 \cdot (5 - 3 \sqrt{2}) (35 - 7 \sqrt{2})})}{483}$

Paso 17.1.2.2

Combina con la regla del producto para radicales.

$\frac{2 (25 \sqrt{3 \cdot (5 - 3 \sqrt{2}) (35 - 7 \sqrt{2})} + 5 \sqrt{6 \cdot (5 - 3 \sqrt{2}) (35 - 7 \sqrt{2})} + 15 \sqrt{3 \cdot (5 - 3 \sqrt{2}) (35 - 7 \sqrt{2}) \cdot 2} + 3 \sqrt{2} \sqrt{6 \cdot (5 - 3 \sqrt{2}) (35 - 7 \sqrt{2})})}{483}$

Paso 17.1.2.3

Multiplica $2$ por $3$ .

$\frac{2 (25 \sqrt{3 \cdot (5 - 3 \sqrt{2}) (35 - 7 \sqrt{2})} + 5 \sqrt{6 \cdot (5 - 3 \sqrt{2}) (35 - 7 \sqrt{2})} + 15 \sqrt{6 \cdot (5 - 3 \sqrt{2}) (35 - 7 \sqrt{2})} + 3 \sqrt{2} \sqrt{6 \cdot (5 - 3 \sqrt{2}) (35 - 7 \sqrt{2})})}{483}$

Paso 17.1.3

Multiplica $3 \sqrt{2} \sqrt{6 \cdot (5 - 3 \sqrt{2}) (35 - 7 \sqrt{2})}$ .

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Paso 17.1.3.1

Combina con la regla del producto para radicales.

$\frac{2 (25 \sqrt{3 \cdot (5 - 3 \sqrt{2}) (35 - 7 \sqrt{2})} + 5 \sqrt{6 \cdot (5 - 3 \sqrt{2}) (35 - 7 \sqrt{2})} + 15 \sqrt{6 \cdot (5 - 3 \sqrt{2}) (35 - 7 \sqrt{2})} + 3 \sqrt{6 \cdot (5 - 3 \sqrt{2}) (35 - 7 \sqrt{2}) \cdot 2})}{483}$

Paso 17.1.3.2

Multiplica $2$ por $6$ .

$\frac{2 (25 \sqrt{3 \cdot (5 - 3 \sqrt{2}) (35 - 7 \sqrt{2})} + 5 \sqrt{6 \cdot (5 - 3 \sqrt{2}) (35 - 7 \sqrt{2})} + 15 \sqrt{6 \cdot (5 - 3 \sqrt{2}) (35 - 7 \sqrt{2})} + 3 \sqrt{12 \cdot (5 - 3 \sqrt{2}) (35 - 7 \sqrt{2})})}{483}$

Paso 17.1.4

Reescribe $12 \cdot (5 - 3 \sqrt{2}) (35 - 7 \sqrt{2})$ como $2^{2} \cdot (3 \cdot (5 - 3 \sqrt{2}) (35 - 7 \sqrt{2}))$ .

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Paso 17.1.4.1

Factoriza $4$ de $12$ .

$\frac{2 (25 \sqrt{3 \cdot (5 - 3 \sqrt{2}) (35 - 7 \sqrt{2})} + 5 \sqrt{6 \cdot (5 - 3 \sqrt{2}) (35 - 7 \sqrt{2})} + 15 \sqrt{6 \cdot (5 - 3 \sqrt{2}) (35 - 7 \sqrt{2})} + 3 \sqrt{4 (3) \cdot (5 - 3 \sqrt{2}) (35 - 7 \sqrt{2})})}{483}$

Paso 17.1.4.2

Reescribe $4$ como $2^{2}$ .

$\frac{2 (25 \sqrt{3 \cdot (5 - 3 \sqrt{2}) (35 - 7 \sqrt{2})} + 5 \sqrt{6 \cdot (5 - 3 \sqrt{2}) (35 - 7 \sqrt{2})} + 15 \sqrt{6 \cdot (5 - 3 \sqrt{2}) (35 - 7 \sqrt{2})} + 3 \sqrt{2^{2} \cdot 3 \cdot (5 - 3 \sqrt{2}) (35 - 7 \sqrt{2})})}{483}$

Paso 17.1.4.3

Agrega paréntesis.

Paso 17.1.4.4

Agrega paréntesis.

$\frac{2 (25 \sqrt{3 \cdot (5 - 3 \sqrt{2}) (35 - 7 \sqrt{2})} + 5 \sqrt{6 \cdot (5 - 3 \sqrt{2}) (35 - 7 \sqrt{2})} + 15 \sqrt{6 \cdot (5 - 3 \sqrt{2}) (35 - 7 \sqrt{2})} + 3 \sqrt{2^{2} \cdot (3 \cdot (5 - 3 \sqrt{2}) (35 - 7 \sqrt{2}))})}{483}$

Paso 17.1.5

Retira los términos de abajo del radical.

$\frac{2 (25 \sqrt{3 \cdot (5 - 3 \sqrt{2}) (35 - 7 \sqrt{2})} + 5 \sqrt{6 \cdot (5 - 3 \sqrt{2}) (35 - 7 \sqrt{2})} + 15 \sqrt{6 \cdot (5 - 3 \sqrt{2}) (35 - 7 \sqrt{2})} + 3 (2 \sqrt{3 \cdot (5 - 3 \sqrt{2}) (35 - 7 \sqrt{2})}))}{483}$

Paso 17.1.6

Multiplica $2$ por $3$ .

$\frac{2 (25 \sqrt{3 \cdot (5 - 3 \sqrt{2}) (35 - 7 \sqrt{2})} + 5 \sqrt{6 \cdot (5 - 3 \sqrt{2}) (35 - 7 \sqrt{2})} + 15 \sqrt{6 \cdot (5 - 3 \sqrt{2}) (35 - 7 \sqrt{2})} + 6 \sqrt{3 \cdot (5 - 3 \sqrt{2}) (35 - 7 \sqrt{2})})}{483}$

Paso 17.2

Suma $25 \sqrt{3 \cdot (5 - 3 \sqrt{2}) (35 - 7 \sqrt{2})}$ y $6 \sqrt{3 \cdot (5 - 3 \sqrt{2}) (35 - 7 \sqrt{2})}$ .

$\frac{2 (31 \sqrt{3 \cdot (5 - 3 \sqrt{2}) (35 - 7 \sqrt{2})} + 5 \sqrt{6 \cdot (5 - 3 \sqrt{2}) (35 - 7 \sqrt{2})} + 15 \sqrt{6 \cdot (5 - 3 \sqrt{2}) (35 - 7 \sqrt{2})})}{483}$

Paso 17.3

Suma $5 \sqrt{6 \cdot (5 - 3 \sqrt{2}) (35 - 7 \sqrt{2})}$ y $15 \sqrt{6 \cdot (5 - 3 \sqrt{2}) (35 - 7 \sqrt{2})}$ .

$\frac{2 (31 \sqrt{3 \cdot (5 - 3 \sqrt{2}) (35 - 7 \sqrt{2})} + 20 \sqrt{6 \cdot (5 - 3 \sqrt{2}) (35 - 7 \sqrt{2})})}{483}$

$\frac{2 (31 \sqrt{3 (5 - 3 \sqrt{2}) (35 - 7 \sqrt{2})} + 20 \sqrt{6 (5 - 3 \sqrt{2}) (35 - 7 \sqrt{2})})}{483}$

Paso 18

El resultado puede mostrarse de distintas formas.

Forma exacta:

$\frac{2 (31 \sqrt{3 (5 - 3 \sqrt{2}) (35 - 7 \sqrt{2})} + 20 \sqrt{6 (5 - 3 \sqrt{2}) (35 - 7 \sqrt{2})})}{483}$

Forma decimal:

$1.85385256 \dots$