Prakalkulus Contoh

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

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

Langkah 1

Gunakan

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

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ untuk menuliskan kembali

\sqrt{x^{2} + 2}

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

{(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}}

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

Langkah 2

Faktorkan

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

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

{(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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Langkah 2.1

Faktorkan

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

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

{(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}}

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

Langkah 2.2

Faktorkan

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

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

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}}

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

Langkah 2.3

Faktorkan

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

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

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}

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

Langkah 2.4

Faktorkan

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

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

{(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}

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

Langkah 2.5

Faktorkan

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

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

{(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})

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

Langkah 3

Bagilah

4

$4$ dengan

2

$2$ .

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

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

Langkah 4

Tulis kembali

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

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

(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})

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

Langkah 5

Perluas

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

$(x^{2} + 2) (x^{2} + 2)$ menggunakan Metode FOIL.

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Langkah 5.1

Terapkan sifat distributif.

{(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})

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

Langkah 5.2

Terapkan sifat distributif.

{(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})

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

Langkah 5.3

Terapkan sifat distributif.

{(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})

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

Langkah 6

Sederhanakan dan gabungkan suku-suku sejenis.

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Langkah 6.1

Sederhanakan setiap suku.

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Langkah 6.1.1

Kalikan

x^{2}

$x^{2}$ dengan

x^{2}

$x^{2}$ dengan menambahkan eksponennya.

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Langkah 6.1.1.1

Gunakan kaidah pangkat

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

$a^{m} a^{n} = a^{m + n}$ untuk menggabungkan pangkat.

{(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})

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

Langkah 6.1.1.2

Tambahkan

2

$2$ dan

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})

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

{(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})

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

Langkah 6.1.2

Pindahkan

2

$2$ ke sebelah kiri

x^{2}

$x^{2}$ .

{(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})

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

Langkah 6.1.3

Kalikan

2

$2$ dengan

2

$2$ .

{(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})

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

{(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})

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

Langkah 6.2

Tambahkan

2 x^{2}

$2 x^{2}$ dan

2 x^{2}

$2 x^{2}$ .

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

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

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

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

Langkah 7

Bagilah

2

$2$ dengan

2

$2$ .

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

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

Langkah 8

Sederhanakan.

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

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

Langkah 9

Terapkan sifat distributif.

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

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

Langkah 10

Kalikan

x

$x$ dengan

x^{2}

$x^{2}$ dengan menambahkan eksponennya.

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Langkah 10.1

Pindahkan

x^{2}

$x^{2}$ .

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

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

Langkah 10.2

Kalikan

x^{2}

$x^{2}$ dengan

x

$x$ .

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Langkah 10.2.1

Naikkan

x

$x$ menjadi pangkat

1

$1$ .

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

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

Langkah 10.2.2

Gunakan kaidah pangkat

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

$a^{m} a^{n} = a^{m + n}$ untuk menggabungkan pangkat.

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

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

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

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

Langkah 10.3

Tambahkan

2

$2$ dan

1

$1$ .

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

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

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

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

Langkah 11

Kalikan

2

$2$ dengan

2

$2$ .

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

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

Langkah 12

Tambahkan

4 x^{2}

$4 x^{2}$ dan

x^{2}

$x^{2}$ .

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

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

Langkah 13

Susun kembali suku-suku.

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

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