Pré-cálculo Exemplos

{(x^{2} + 2)}^{\frac{5}{2}} + 2 x {(x^{2} + 2)}^{\frac{3}{2}} + x^{2} \sqrt{x^{2} + 2}

${(x^{2} + 2)}^{\frac{5}{2}} + 2 x {(x^{2} + 2)}^{\frac{3}{2}} + x^{2} \sqrt{x^{2} + 2}$

Etapa 1

Use

\sqrt[n]{a^{x}} = a^{\frac{x}{n}}

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ para reescrever

\sqrt{x^{2} + 2}

$\sqrt{x^{2} + 2}$ como

{(x^{2} + 2)}^{\frac{1}{2}}

${(x^{2} + 2)}^{\frac{1}{2}}$ .

{(x^{2} + 2)}^{\frac{5}{2}} + 2 x {(x^{2} + 2)}^{\frac{3}{2}} + x^{2} {(x^{2} + 2)}^{\frac{1}{2}}

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

Etapa 2

Fatore

{(x^{2} + 2)}^{\frac{1}{2}}

${(x^{2} + 2)}^{\frac{1}{2}}$ de

{(x^{2} + 2)}^{\frac{5}{2}} + 2 x {(x^{2} + 2)}^{\frac{3}{2}} + x^{2} {(x^{2} + 2)}^{\frac{1}{2}}

${(x^{2} + 2)}^{\frac{5}{2}} + 2 x {(x^{2} + 2)}^{\frac{3}{2}} + x^{2} {(x^{2} + 2)}^{\frac{1}{2}}$ .

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Etapa 2.1

Fatore

{(x^{2} + 2)}^{\frac{1}{2}}

${(x^{2} + 2)}^{\frac{1}{2}}$ de

{(x^{2} + 2)}^{\frac{5}{2}}

${(x^{2} + 2)}^{\frac{5}{2}}$ .

{(x^{2} + 2)}^{\frac{1}{2}} {(x^{2} + 2)}^{\frac{4}{2}} + 2 x {(x^{2} + 2)}^{\frac{3}{2}} + x^{2} {(x^{2} + 2)}^{\frac{1}{2}}

${(x^{2} + 2)}^{\frac{1}{2}} {(x^{2} + 2)}^{\frac{4}{2}} + 2 x {(x^{2} + 2)}^{\frac{3}{2}} + x^{2} {(x^{2} + 2)}^{\frac{1}{2}}$

Etapa 2.2

Fatore

{(x^{2} + 2)}^{\frac{1}{2}}

${(x^{2} + 2)}^{\frac{1}{2}}$ de

2 x {(x^{2} + 2)}^{\frac{3}{2}}

$2 x {(x^{2} + 2)}^{\frac{3}{2}}$ .

{(x^{2} + 2)}^{\frac{1}{2}} {(x^{2} + 2)}^{\frac{4}{2}} + {(x^{2} + 2)}^{\frac{1}{2}} (2 x {(x^{2} + 2)}^{\frac{2}{2}}) + x^{2} {(x^{2} + 2)}^{\frac{1}{2}}

${(x^{2} + 2)}^{\frac{1}{2}} {(x^{2} + 2)}^{\frac{4}{2}} + {(x^{2} + 2)}^{\frac{1}{2}} (2 x {(x^{2} + 2)}^{\frac{2}{2}}) + x^{2} {(x^{2} + 2)}^{\frac{1}{2}}$

Etapa 2.3

Fatore

{(x^{2} + 2)}^{\frac{1}{2}}

${(x^{2} + 2)}^{\frac{1}{2}}$ de

x^{2} {(x^{2} + 2)}^{\frac{1}{2}}

$x^{2} {(x^{2} + 2)}^{\frac{1}{2}}$ .

{(x^{2} + 2)}^{\frac{1}{2}} {(x^{2} + 2)}^{\frac{4}{2}} + {(x^{2} + 2)}^{\frac{1}{2}} (2 x {(x^{2} + 2)}^{\frac{2}{2}}) + {(x^{2} + 2)}^{\frac{1}{2}} x^{2}

${(x^{2} + 2)}^{\frac{1}{2}} {(x^{2} + 2)}^{\frac{4}{2}} + {(x^{2} + 2)}^{\frac{1}{2}} (2 x {(x^{2} + 2)}^{\frac{2}{2}}) + {(x^{2} + 2)}^{\frac{1}{2}} x^{2}$

Etapa 2.4

Fatore

{(x^{2} + 2)}^{\frac{1}{2}}

${(x^{2} + 2)}^{\frac{1}{2}}$ de

{(x^{2} + 2)}^{\frac{1}{2}} {(x^{2} + 2)}^{\frac{4}{2}} + {(x^{2} + 2)}^{\frac{1}{2}} (2 x {(x^{2} + 2)}^{\frac{2}{2}})

${(x^{2} + 2)}^{\frac{1}{2}} {(x^{2} + 2)}^{\frac{4}{2}} + {(x^{2} + 2)}^{\frac{1}{2}} (2 x {(x^{2} + 2)}^{\frac{2}{2}})$ .

{(x^{2} + 2)}^{\frac{1}{2}} ({(x^{2} + 2)}^{\frac{4}{2}} + 2 x {(x^{2} + 2)}^{\frac{2}{2}}) + {(x^{2} + 2)}^{\frac{1}{2}} x^{2}

${(x^{2} + 2)}^{\frac{1}{2}} ({(x^{2} + 2)}^{\frac{4}{2}} + 2 x {(x^{2} + 2)}^{\frac{2}{2}}) + {(x^{2} + 2)}^{\frac{1}{2}} x^{2}$

Etapa 2.5

Fatore

{(x^{2} + 2)}^{\frac{1}{2}}

${(x^{2} + 2)}^{\frac{1}{2}}$ de

{(x^{2} + 2)}^{\frac{1}{2}} ({(x^{2} + 2)}^{\frac{4}{2}} + 2 x {(x^{2} + 2)}^{\frac{2}{2}}) + {(x^{2} + 2)}^{\frac{1}{2}} x^{2}

${(x^{2} + 2)}^{\frac{1}{2}} ({(x^{2} + 2)}^{\frac{4}{2}} + 2 x {(x^{2} + 2)}^{\frac{2}{2}}) + {(x^{2} + 2)}^{\frac{1}{2}} x^{2}$ .

{(x^{2} + 2)}^{\frac{1}{2}} ({(x^{2} + 2)}^{\frac{4}{2}} + 2 x {(x^{2} + 2)}^{\frac{2}{2}} + x^{2})

${(x^{2} + 2)}^{\frac{1}{2}} ({(x^{2} + 2)}^{\frac{4}{2}} + 2 x {(x^{2} + 2)}^{\frac{2}{2}} + x^{2})$

{(x^{2} + 2)}^{\frac{1}{2}} ({(x^{2} + 2)}^{\frac{4}{2}} + 2 x {(x^{2} + 2)}^{\frac{2}{2}} + x^{2})

${(x^{2} + 2)}^{\frac{1}{2}} ({(x^{2} + 2)}^{\frac{4}{2}} + 2 x {(x^{2} + 2)}^{\frac{2}{2}} + x^{2})$

Etapa 3

Divida

4

$4$ por

2

$2$ .

{(x^{2} + 2)}^{\frac{1}{2}} ({(x^{2} + 2)}^{2} + 2 x {(x^{2} + 2)}^{\frac{2}{2}} + x^{2})

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

Etapa 4

Reescreva

{(x^{2} + 2)}^{2}

${(x^{2} + 2)}^{2}$ como

(x^{2} + 2) (x^{2} + 2)

$(x^{2} + 2) (x^{2} + 2)$ .

{(x^{2} + 2)}^{\frac{1}{2}} ((x^{2} + 2) (x^{2} + 2) + 2 x {(x^{2} + 2)}^{\frac{2}{2}} + x^{2})

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

Etapa 5

Expanda

(x^{2} + 2) (x^{2} + 2)

$(x^{2} + 2) (x^{2} + 2)$ usando o método FOIL.

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Etapa 5.1

Aplique a propriedade distributiva.

{(x^{2} + 2)}^{\frac{1}{2}} (x^{2} (x^{2} + 2) + 2 (x^{2} + 2) + 2 x {(x^{2} + 2)}^{\frac{2}{2}} + x^{2})

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

Etapa 5.2

Aplique a propriedade distributiva.

{(x^{2} + 2)}^{\frac{1}{2}} (x^{2} x^{2} + x^{2} \cdot 2 + 2 (x^{2} + 2) + 2 x {(x^{2} + 2)}^{\frac{2}{2}} + x^{2})

${(x^{2} + 2)}^{\frac{1}{2}} (x^{2} x^{2} + x^{2} \cdot 2 + 2 (x^{2} + 2) + 2 x {(x^{2} + 2)}^{\frac{2}{2}} + x^{2})$

Etapa 5.3

Aplique a propriedade distributiva.

{(x^{2} + 2)}^{\frac{1}{2}} (x^{2} x^{2} + x^{2} \cdot 2 + 2 x^{2} + 2 \cdot 2 + 2 x {(x^{2} + 2)}^{\frac{2}{2}} + x^{2})

${(x^{2} + 2)}^{\frac{1}{2}} (x^{2} x^{2} + x^{2} \cdot 2 + 2 x^{2} + 2 \cdot 2 + 2 x {(x^{2} + 2)}^{\frac{2}{2}} + x^{2})$

{(x^{2} + 2)}^{\frac{1}{2}} (x^{2} x^{2} + x^{2} \cdot 2 + 2 x^{2} + 2 \cdot 2 + 2 x {(x^{2} + 2)}^{\frac{2}{2}} + x^{2})

${(x^{2} + 2)}^{\frac{1}{2}} (x^{2} x^{2} + x^{2} \cdot 2 + 2 x^{2} + 2 \cdot 2 + 2 x {(x^{2} + 2)}^{\frac{2}{2}} + x^{2})$

Etapa 6

Simplifique e combine termos semelhantes.

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Etapa 6.1

Simplifique cada termo.

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Etapa 6.1.1

Multiplique

x^{2}

$x^{2}$ por

x^{2}

$x^{2}$ somando os expoentes.

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Etapa 6.1.1.1

Use a regra da multiplicação de potências

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ para combinar expoentes.

{(x^{2} + 2)}^{\frac{1}{2}} (x^{2 + 2} + x^{2} \cdot 2 + 2 x^{2} + 2 \cdot 2 + 2 x {(x^{2} + 2)}^{\frac{2}{2}} + x^{2})

${(x^{2} + 2)}^{\frac{1}{2}} (x^{2 + 2} + x^{2} \cdot 2 + 2 x^{2} + 2 \cdot 2 + 2 x {(x^{2} + 2)}^{\frac{2}{2}} + x^{2})$

Etapa 6.1.1.2

Some

2

$2$ e

2

$2$ .

{(x^{2} + 2)}^{\frac{1}{2}} (x^{4} + x^{2} \cdot 2 + 2 x^{2} + 2 \cdot 2 + 2 x {(x^{2} + 2)}^{\frac{2}{2}} + x^{2})

${(x^{2} + 2)}^{\frac{1}{2}} (x^{4} + x^{2} \cdot 2 + 2 x^{2} + 2 \cdot 2 + 2 x {(x^{2} + 2)}^{\frac{2}{2}} + x^{2})$

Etapa 6.1.2

Mova

$2$ para a esquerda de

$x^{2}$ .

${(x^{2} + 2)}^{\frac{1}{2}} (x^{4} + 2 x^{2} + 2 x^{2} + 2 \cdot 2 + 2 x {(x^{2} + 2)}^{\frac{2}{2}} + x^{2})$

Etapa 6.1.3

Multiplique

$2$ por

$2$ .

${(x^{2} + 2)}^{\frac{1}{2}} (x^{4} + 2 x^{2} + 2 x^{2} + 4 + 2 x {(x^{2} + 2)}^{\frac{2}{2}} + x^{2})$

Etapa 6.2

Some

$2 x^{2}$ e

$2 x^{2}$ .

${(x^{2} + 2)}^{\frac{1}{2}} (x^{4} + 4 x^{2} + 4 + 2 x {(x^{2} + 2)}^{\frac{2}{2}} + x^{2})$

Etapa 7

Divida

$2$ por

$2$ .

${(x^{2} + 2)}^{\frac{1}{2}} (x^{4} + 4 x^{2} + 4 + 2 x {(x^{2} + 2)}^{1} + x^{2})$

Etapa 8

Simplifique.

${(x^{2} + 2)}^{\frac{1}{2}} (x^{4} + 4 x^{2} + 4 + 2 x (x^{2} + 2) + x^{2})$

Etapa 9

Aplique a propriedade distributiva.

${(x^{2} + 2)}^{\frac{1}{2}} (x^{4} + 4 x^{2} + 4 + 2 x \cdot x^{2} + 2 x \cdot 2 + x^{2})$

Etapa 10

Multiplique

$x$ por

$x^{2}$ somando os expoentes.

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Etapa 10.1

Mova

$x^{2}$ .

${(x^{2} + 2)}^{\frac{1}{2}} (x^{4} + 4 x^{2} + 4 + 2 (x^{2} x) + 2 x \cdot 2 + x^{2})$

Etapa 10.2

Multiplique

$x^{2}$ por

$x$ .

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Etapa 10.2.1

Eleve

$x$ à potência de

$1$ .

${(x^{2} + 2)}^{\frac{1}{2}} (x^{4} + 4 x^{2} + 4 + 2 (x^{2} x^{1}) + 2 x \cdot 2 + x^{2})$

Etapa 10.2.2

Use a regra da multiplicação de potências

$a^{m} a^{n} = a^{m + n}$ para combinar expoentes.

${(x^{2} + 2)}^{\frac{1}{2}} (x^{4} + 4 x^{2} + 4 + 2 x^{2 + 1} + 2 x \cdot 2 + x^{2})$

Etapa 10.3

Some

$2$ e

$1$ .

${(x^{2} + 2)}^{\frac{1}{2}} (x^{4} + 4 x^{2} + 4 + 2 x^{3} + 2 x \cdot 2 + x^{2})$

Etapa 11

Multiplique

$2$ por

$2$ .

${(x^{2} + 2)}^{\frac{1}{2}} (x^{4} + 4 x^{2} + 4 + 2 x^{3} + 4 x + x^{2})$

Etapa 12

Some

$4 x^{2}$ e

$x^{2}$ .

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

Etapa 13

Reordene os termos.

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