Lineare Algebra Beispiele

$[\begin{array}{c} \frac{2}{7} & \frac{5}{7} & \frac{2}{7} & - \frac{4}{7} \\ \frac{1}{\sqrt{22}} & \frac{2}{\sqrt{22}} & - \frac{4}{\sqrt{22}} & \frac{1}{\sqrt{22}} \end{array}] [\begin{array}{c} 2 & 3 \\ 5 & 7 \\ 2 & - 2 \\ - 4 & - 3 \end{array}]$

Schritt 1

Mutltipliziere $\frac{1}{\sqrt{22}}$ mit $\frac{\sqrt{22}}{\sqrt{22}}$ .

$[\begin{array}{c} \frac{2}{7} & \frac{5}{7} & \frac{2}{7} & - \frac{4}{7} \\ \frac{1}{\sqrt{22}} \cdot \frac{\sqrt{22}}{\sqrt{22}} & \frac{2}{\sqrt{22}} & - \frac{4}{\sqrt{22}} & \frac{1}{\sqrt{22}} \end{array}] [\begin{array}{c} 2 & 3 \\ 5 & 7 \\ 2 & - 2 \\ - 4 & - 3 \end{array}]$

Schritt 2

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Schritt 2.1

Mutltipliziere $\frac{1}{\sqrt{22}}$ mit $\frac{\sqrt{22}}{\sqrt{22}}$ .

$[\begin{array}{c} \frac{2}{7} & \frac{5}{7} & \frac{2}{7} & - \frac{4}{7} \\ \frac{\sqrt{22}}{\sqrt{22} \sqrt{22}} & \frac{2}{\sqrt{22}} & - \frac{4}{\sqrt{22}} & \frac{1}{\sqrt{22}} \end{array}] [\begin{array}{c} 2 & 3 \\ 5 & 7 \\ 2 & - 2 \\ - 4 & - 3 \end{array}]$

Schritt 2.2

Potenziere $\sqrt{22}$ mit $1$ .

$[\begin{array}{c} \frac{2}{7} & \frac{5}{7} & \frac{2}{7} & - \frac{4}{7} \\ \frac{\sqrt{22}}{{\sqrt{22}}^{1} \sqrt{22}} & \frac{2}{\sqrt{22}} & - \frac{4}{\sqrt{22}} & \frac{1}{\sqrt{22}} \end{array}] [\begin{array}{c} 2 & 3 \\ 5 & 7 \\ 2 & - 2 \\ - 4 & - 3 \end{array}]$

Schritt 2.3

Potenziere $\sqrt{22}$ mit $1$ .

$[\begin{array}{c} \frac{2}{7} & \frac{5}{7} & \frac{2}{7} & - \frac{4}{7} \\ \frac{\sqrt{22}}{{\sqrt{22}}^{1} {\sqrt{22}}^{1}} & \frac{2}{\sqrt{22}} & - \frac{4}{\sqrt{22}} & \frac{1}{\sqrt{22}} \end{array}] [\begin{array}{c} 2 & 3 \\ 5 & 7 \\ 2 & - 2 \\ - 4 & - 3 \end{array}]$

Schritt 2.4

Wende die Exponentenregel $a^{m} a^{n} = a^{m + n}$ an, um die Exponenten zu kombinieren.

$[\begin{array}{c} \frac{2}{7} & \frac{5}{7} & \frac{2}{7} & - \frac{4}{7} \\ \frac{\sqrt{22}}{{\sqrt{22}}^{1 + 1}} & \frac{2}{\sqrt{22}} & - \frac{4}{\sqrt{22}} & \frac{1}{\sqrt{22}} \end{array}] [\begin{array}{c} 2 & 3 \\ 5 & 7 \\ 2 & - 2 \\ - 4 & - 3 \end{array}]$

Schritt 2.5

Addiere $1$ und $1$ .

$[\begin{array}{c} \frac{2}{7} & \frac{5}{7} & \frac{2}{7} & - \frac{4}{7} \\ \frac{\sqrt{22}}{{\sqrt{22}}^{2}} & \frac{2}{\sqrt{22}} & - \frac{4}{\sqrt{22}} & \frac{1}{\sqrt{22}} \end{array}] [\begin{array}{c} 2 & 3 \\ 5 & 7 \\ 2 & - 2 \\ - 4 & - 3 \end{array}]$

Schritt 2.6

Schreibe ${\sqrt{22}}^{2}$ als $22$ um.

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Schritt 2.6.1

Benutze $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ , um $\sqrt{22}$ als $22^{\frac{1}{2}}$ neu zu schreiben.

$[\begin{array}{c} \frac{2}{7} & \frac{5}{7} & \frac{2}{7} & - \frac{4}{7} \\ \frac{\sqrt{22}}{{(22^{\frac{1}{2}})}^{2}} & \frac{2}{\sqrt{22}} & - \frac{4}{\sqrt{22}} & \frac{1}{\sqrt{22}} \end{array}] [\begin{array}{c} 2 & 3 \\ 5 & 7 \\ 2 & - 2 \\ - 4 & - 3 \end{array}]$

Schritt 2.6.2

Wende die Potenzregel an und multipliziere die Exponenten, ${(a^{m})}^{n} = a^{m n}$ .

$[\begin{array}{c} \frac{2}{7} & \frac{5}{7} & \frac{2}{7} & - \frac{4}{7} \\ \frac{\sqrt{22}}{22^{\frac{1}{2} \cdot 2}} & \frac{2}{\sqrt{22}} & - \frac{4}{\sqrt{22}} & \frac{1}{\sqrt{22}} \end{array}] [\begin{array}{c} 2 & 3 \\ 5 & 7 \\ 2 & - 2 \\ - 4 & - 3 \end{array}]$

Schritt 2.6.3

Kombiniere $\frac{1}{2}$ und $2$ .

$[\begin{array}{c} \frac{2}{7} & \frac{5}{7} & \frac{2}{7} & - \frac{4}{7} \\ \frac{\sqrt{22}}{22^{\frac{2}{2}}} & \frac{2}{\sqrt{22}} & - \frac{4}{\sqrt{22}} & \frac{1}{\sqrt{22}} \end{array}] [\begin{array}{c} 2 & 3 \\ 5 & 7 \\ 2 & - 2 \\ - 4 & - 3 \end{array}]$

Schritt 2.6.4

Kürze den gemeinsamen Faktor von $2$ .

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Schritt 2.6.4.1

Kürze den gemeinsamen Faktor.

Schritt 2.6.4.2

Forme den Ausdruck um.

$[\begin{array}{c} \frac{2}{7} & \frac{5}{7} & \frac{2}{7} & - \frac{4}{7} \\ \frac{\sqrt{22}}{22^{1}} & \frac{2}{\sqrt{22}} & - \frac{4}{\sqrt{22}} & \frac{1}{\sqrt{22}} \end{array}] [\begin{array}{c} 2 & 3 \\ 5 & 7 \\ 2 & - 2 \\ - 4 & - 3 \end{array}]$

Schritt 2.6.5

Berechne den Exponenten.

$[\begin{array}{c} \frac{2}{7} & \frac{5}{7} & \frac{2}{7} & - \frac{4}{7} \\ \frac{\sqrt{22}}{22} & \frac{2}{\sqrt{22}} & - \frac{4}{\sqrt{22}} & \frac{1}{\sqrt{22}} \end{array}] [\begin{array}{c} 2 & 3 \\ 5 & 7 \\ 2 & - 2 \\ - 4 & - 3 \end{array}]$

Schritt 3

Mutltipliziere $\frac{2}{\sqrt{22}}$ mit $\frac{\sqrt{22}}{\sqrt{22}}$ .

$[\begin{array}{c} \frac{2}{7} & \frac{5}{7} & \frac{2}{7} & - \frac{4}{7} \\ \frac{\sqrt{22}}{22} & \frac{2}{\sqrt{22}} \cdot \frac{\sqrt{22}}{\sqrt{22}} & - \frac{4}{\sqrt{22}} & \frac{1}{\sqrt{22}} \end{array}] [\begin{array}{c} 2 & 3 \\ 5 & 7 \\ 2 & - 2 \\ - 4 & - 3 \end{array}]$

Schritt 4

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Schritt 4.1

Mutltipliziere $\frac{2}{\sqrt{22}}$ mit $\frac{\sqrt{22}}{\sqrt{22}}$ .

$[\begin{array}{c} \frac{2}{7} & \frac{5}{7} & \frac{2}{7} & - \frac{4}{7} \\ \frac{\sqrt{22}}{22} & \frac{2 \sqrt{22}}{\sqrt{22} \sqrt{22}} & - \frac{4}{\sqrt{22}} & \frac{1}{\sqrt{22}} \end{array}] [\begin{array}{c} 2 & 3 \\ 5 & 7 \\ 2 & - 2 \\ - 4 & - 3 \end{array}]$

Schritt 4.2

Potenziere $\sqrt{22}$ mit $1$ .

$[\begin{array}{c} \frac{2}{7} & \frac{5}{7} & \frac{2}{7} & - \frac{4}{7} \\ \frac{\sqrt{22}}{22} & \frac{2 \sqrt{22}}{{\sqrt{22}}^{1} \sqrt{22}} & - \frac{4}{\sqrt{22}} & \frac{1}{\sqrt{22}} \end{array}] [\begin{array}{c} 2 & 3 \\ 5 & 7 \\ 2 & - 2 \\ - 4 & - 3 \end{array}]$

Schritt 4.3

Potenziere $\sqrt{22}$ mit $1$ .

$[\begin{array}{c} \frac{2}{7} & \frac{5}{7} & \frac{2}{7} & - \frac{4}{7} \\ \frac{\sqrt{22}}{22} & \frac{2 \sqrt{22}}{{\sqrt{22}}^{1} {\sqrt{22}}^{1}} & - \frac{4}{\sqrt{22}} & \frac{1}{\sqrt{22}} \end{array}] [\begin{array}{c} 2 & 3 \\ 5 & 7 \\ 2 & - 2 \\ - 4 & - 3 \end{array}]$

Schritt 4.4

Wende die Exponentenregel $a^{m} a^{n} = a^{m + n}$ an, um die Exponenten zu kombinieren.

$[\begin{array}{c} \frac{2}{7} & \frac{5}{7} & \frac{2}{7} & - \frac{4}{7} \\ \frac{\sqrt{22}}{22} & \frac{2 \sqrt{22}}{{\sqrt{22}}^{1 + 1}} & - \frac{4}{\sqrt{22}} & \frac{1}{\sqrt{22}} \end{array}] [\begin{array}{c} 2 & 3 \\ 5 & 7 \\ 2 & - 2 \\ - 4 & - 3 \end{array}]$

Schritt 4.5

Addiere $1$ und $1$ .

$[\begin{array}{c} \frac{2}{7} & \frac{5}{7} & \frac{2}{7} & - \frac{4}{7} \\ \frac{\sqrt{22}}{22} & \frac{2 \sqrt{22}}{{\sqrt{22}}^{2}} & - \frac{4}{\sqrt{22}} & \frac{1}{\sqrt{22}} \end{array}] [\begin{array}{c} 2 & 3 \\ 5 & 7 \\ 2 & - 2 \\ - 4 & - 3 \end{array}]$

Schritt 4.6

Schreibe ${\sqrt{22}}^{2}$ als $22$ um.

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Schritt 4.6.1

Benutze $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ , um $\sqrt{22}$ als $22^{\frac{1}{2}}$ neu zu schreiben.

$[\begin{array}{c} \frac{2}{7} & \frac{5}{7} & \frac{2}{7} & - \frac{4}{7} \\ \frac{\sqrt{22}}{22} & \frac{2 \sqrt{22}}{{(22^{\frac{1}{2}})}^{2}} & - \frac{4}{\sqrt{22}} & \frac{1}{\sqrt{22}} \end{array}] [\begin{array}{c} 2 & 3 \\ 5 & 7 \\ 2 & - 2 \\ - 4 & - 3 \end{array}]$

Schritt 4.6.2

Wende die Potenzregel an und multipliziere die Exponenten, ${(a^{m})}^{n} = a^{m n}$ .

$[\begin{array}{c} \frac{2}{7} & \frac{5}{7} & \frac{2}{7} & - \frac{4}{7} \\ \frac{\sqrt{22}}{22} & \frac{2 \sqrt{22}}{22^{\frac{1}{2} \cdot 2}} & - \frac{4}{\sqrt{22}} & \frac{1}{\sqrt{22}} \end{array}] [\begin{array}{c} 2 & 3 \\ 5 & 7 \\ 2 & - 2 \\ - 4 & - 3 \end{array}]$

Schritt 4.6.3

Kombiniere $\frac{1}{2}$ und $2$ .

$[\begin{array}{c} \frac{2}{7} & \frac{5}{7} & \frac{2}{7} & - \frac{4}{7} \\ \frac{\sqrt{22}}{22} & \frac{2 \sqrt{22}}{22^{\frac{2}{2}}} & - \frac{4}{\sqrt{22}} & \frac{1}{\sqrt{22}} \end{array}] [\begin{array}{c} 2 & 3 \\ 5 & 7 \\ 2 & - 2 \\ - 4 & - 3 \end{array}]$

Schritt 4.6.4

Kürze den gemeinsamen Faktor von $2$ .

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Schritt 4.6.4.1

Kürze den gemeinsamen Faktor.

Schritt 4.6.4.2

Forme den Ausdruck um.

$[\begin{array}{c} \frac{2}{7} & \frac{5}{7} & \frac{2}{7} & - \frac{4}{7} \\ \frac{\sqrt{22}}{22} & \frac{2 \sqrt{22}}{22^{1}} & - \frac{4}{\sqrt{22}} & \frac{1}{\sqrt{22}} \end{array}] [\begin{array}{c} 2 & 3 \\ 5 & 7 \\ 2 & - 2 \\ - 4 & - 3 \end{array}]$

Schritt 4.6.5

Berechne den Exponenten.

$[\begin{array}{c} \frac{2}{7} & \frac{5}{7} & \frac{2}{7} & - \frac{4}{7} \\ \frac{\sqrt{22}}{22} & \frac{2 \sqrt{22}}{22} & - \frac{4}{\sqrt{22}} & \frac{1}{\sqrt{22}} \end{array}] [\begin{array}{c} 2 & 3 \\ 5 & 7 \\ 2 & - 2 \\ - 4 & - 3 \end{array}]$

Schritt 5

Kürze den gemeinsamen Teiler von $2$ und $22$ .

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Schritt 5.1

Faktorisiere $2$ aus $2 \sqrt{22}$ heraus.

$[\begin{array}{c} \frac{2}{7} & \frac{5}{7} & \frac{2}{7} & - \frac{4}{7} \\ \frac{\sqrt{22}}{22} & \frac{2 (\sqrt{22})}{22} & - \frac{4}{\sqrt{22}} & \frac{1}{\sqrt{22}} \end{array}] [\begin{array}{c} 2 & 3 \\ 5 & 7 \\ 2 & - 2 \\ - 4 & - 3 \end{array}]$

Schritt 5.2

Kürze die gemeinsamen Faktoren.

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Schritt 5.2.1

Faktorisiere $2$ aus $22$ heraus.

$[\begin{array}{c} \frac{2}{7} & \frac{5}{7} & \frac{2}{7} & - \frac{4}{7} \\ \frac{\sqrt{22}}{22} & \frac{2 \sqrt{22}}{2 \cdot 11} & - \frac{4}{\sqrt{22}} & \frac{1}{\sqrt{22}} \end{array}] [\begin{array}{c} 2 & 3 \\ 5 & 7 \\ 2 & - 2 \\ - 4 & - 3 \end{array}]$

Schritt 5.2.2

Kürze den gemeinsamen Faktor.

Schritt 5.2.3

Forme den Ausdruck um.

$[\begin{array}{c} \frac{2}{7} & \frac{5}{7} & \frac{2}{7} & - \frac{4}{7} \\ \frac{\sqrt{22}}{22} & \frac{\sqrt{22}}{11} & - \frac{4}{\sqrt{22}} & \frac{1}{\sqrt{22}} \end{array}] [\begin{array}{c} 2 & 3 \\ 5 & 7 \\ 2 & - 2 \\ - 4 & - 3 \end{array}]$

Schritt 6

Mutltipliziere $\frac{4}{\sqrt{22}}$ mit $\frac{\sqrt{22}}{\sqrt{22}}$ .

$[\begin{array}{c} \frac{2}{7} & \frac{5}{7} & \frac{2}{7} & - \frac{4}{7} \\ \frac{\sqrt{22}}{22} & \frac{\sqrt{22}}{11} & - (\frac{4}{\sqrt{22}} \cdot \frac{\sqrt{22}}{\sqrt{22}}) & \frac{1}{\sqrt{22}} \end{array}] [\begin{array}{c} 2 & 3 \\ 5 & 7 \\ 2 & - 2 \\ - 4 & - 3 \end{array}]$

Schritt 7

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Schritt 7.1

Mutltipliziere $\frac{4}{\sqrt{22}}$ mit $\frac{\sqrt{22}}{\sqrt{22}}$ .

$[\begin{array}{c} \frac{2}{7} & \frac{5}{7} & \frac{2}{7} & - \frac{4}{7} \\ \frac{\sqrt{22}}{22} & \frac{\sqrt{22}}{11} & - \frac{4 \sqrt{22}}{\sqrt{22} \sqrt{22}} & \frac{1}{\sqrt{22}} \end{array}] [\begin{array}{c} 2 & 3 \\ 5 & 7 \\ 2 & - 2 \\ - 4 & - 3 \end{array}]$

Schritt 7.2

Potenziere $\sqrt{22}$ mit $1$ .

$[\begin{array}{c} \frac{2}{7} & \frac{5}{7} & \frac{2}{7} & - \frac{4}{7} \\ \frac{\sqrt{22}}{22} & \frac{\sqrt{22}}{11} & - \frac{4 \sqrt{22}}{{\sqrt{22}}^{1} \sqrt{22}} & \frac{1}{\sqrt{22}} \end{array}] [\begin{array}{c} 2 & 3 \\ 5 & 7 \\ 2 & - 2 \\ - 4 & - 3 \end{array}]$

Schritt 7.3

Potenziere $\sqrt{22}$ mit $1$ .

$[\begin{array}{c} \frac{2}{7} & \frac{5}{7} & \frac{2}{7} & - \frac{4}{7} \\ \frac{\sqrt{22}}{22} & \frac{\sqrt{22}}{11} & - \frac{4 \sqrt{22}}{{\sqrt{22}}^{1} {\sqrt{22}}^{1}} & \frac{1}{\sqrt{22}} \end{array}] [\begin{array}{c} 2 & 3 \\ 5 & 7 \\ 2 & - 2 \\ - 4 & - 3 \end{array}]$

Schritt 7.4

Wende die Exponentenregel $a^{m} a^{n} = a^{m + n}$ an, um die Exponenten zu kombinieren.

$[\begin{array}{c} \frac{2}{7} & \frac{5}{7} & \frac{2}{7} & - \frac{4}{7} \\ \frac{\sqrt{22}}{22} & \frac{\sqrt{22}}{11} & - \frac{4 \sqrt{22}}{{\sqrt{22}}^{1 + 1}} & \frac{1}{\sqrt{22}} \end{array}] [\begin{array}{c} 2 & 3 \\ 5 & 7 \\ 2 & - 2 \\ - 4 & - 3 \end{array}]$

Schritt 7.5

Addiere $1$ und $1$ .

$[\begin{array}{c} \frac{2}{7} & \frac{5}{7} & \frac{2}{7} & - \frac{4}{7} \\ \frac{\sqrt{22}}{22} & \frac{\sqrt{22}}{11} & - \frac{4 \sqrt{22}}{{\sqrt{22}}^{2}} & \frac{1}{\sqrt{22}} \end{array}] [\begin{array}{c} 2 & 3 \\ 5 & 7 \\ 2 & - 2 \\ - 4 & - 3 \end{array}]$

Schritt 7.6

Schreibe ${\sqrt{22}}^{2}$ als $22$ um.

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Schritt 7.6.1

Benutze $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ , um $\sqrt{22}$ als $22^{\frac{1}{2}}$ neu zu schreiben.

$[\begin{array}{c} \frac{2}{7} & \frac{5}{7} & \frac{2}{7} & - \frac{4}{7} \\ \frac{\sqrt{22}}{22} & \frac{\sqrt{22}}{11} & - \frac{4 \sqrt{22}}{{(22^{\frac{1}{2}})}^{2}} & \frac{1}{\sqrt{22}} \end{array}] [\begin{array}{c} 2 & 3 \\ 5 & 7 \\ 2 & - 2 \\ - 4 & - 3 \end{array}]$

Schritt 7.6.2

Wende die Potenzregel an und multipliziere die Exponenten, ${(a^{m})}^{n} = a^{m n}$ .

$[\begin{array}{c} \frac{2}{7} & \frac{5}{7} & \frac{2}{7} & - \frac{4}{7} \\ \frac{\sqrt{22}}{22} & \frac{\sqrt{22}}{11} & - \frac{4 \sqrt{22}}{22^{\frac{1}{2} \cdot 2}} & \frac{1}{\sqrt{22}} \end{array}] [\begin{array}{c} 2 & 3 \\ 5 & 7 \\ 2 & - 2 \\ - 4 & - 3 \end{array}]$

Schritt 7.6.3

Kombiniere $\frac{1}{2}$ und $2$ .

$[\begin{array}{c} \frac{2}{7} & \frac{5}{7} & \frac{2}{7} & - \frac{4}{7} \\ \frac{\sqrt{22}}{22} & \frac{\sqrt{22}}{11} & - \frac{4 \sqrt{22}}{22^{\frac{2}{2}}} & \frac{1}{\sqrt{22}} \end{array}] [\begin{array}{c} 2 & 3 \\ 5 & 7 \\ 2 & - 2 \\ - 4 & - 3 \end{array}]$

Schritt 7.6.4

Kürze den gemeinsamen Faktor von $2$ .

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Schritt 7.6.4.1

Kürze den gemeinsamen Faktor.

Schritt 7.6.4.2

Forme den Ausdruck um.

$[\begin{array}{c} \frac{2}{7} & \frac{5}{7} & \frac{2}{7} & - \frac{4}{7} \\ \frac{\sqrt{22}}{22} & \frac{\sqrt{22}}{11} & - \frac{4 \sqrt{22}}{22^{1}} & \frac{1}{\sqrt{22}} \end{array}] [\begin{array}{c} 2 & 3 \\ 5 & 7 \\ 2 & - 2 \\ - 4 & - 3 \end{array}]$

Schritt 7.6.5

Berechne den Exponenten.

$[\begin{array}{c} \frac{2}{7} & \frac{5}{7} & \frac{2}{7} & - \frac{4}{7} \\ \frac{\sqrt{22}}{22} & \frac{\sqrt{22}}{11} & - \frac{4 \sqrt{22}}{22} & \frac{1}{\sqrt{22}} \end{array}] [\begin{array}{c} 2 & 3 \\ 5 & 7 \\ 2 & - 2 \\ - 4 & - 3 \end{array}]$

Schritt 8

Kürze den gemeinsamen Teiler von $4$ und $22$ .

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Schritt 8.1

Faktorisiere $2$ aus $4 \sqrt{22}$ heraus.

$[\begin{array}{c} \frac{2}{7} & \frac{5}{7} & \frac{2}{7} & - \frac{4}{7} \\ \frac{\sqrt{22}}{22} & \frac{\sqrt{22}}{11} & - \frac{2 (2 \sqrt{22})}{22} & \frac{1}{\sqrt{22}} \end{array}] [\begin{array}{c} 2 & 3 \\ 5 & 7 \\ 2 & - 2 \\ - 4 & - 3 \end{array}]$

Schritt 8.2

Kürze die gemeinsamen Faktoren.

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Schritt 8.2.1

Faktorisiere $2$ aus $22$ heraus.

$[\begin{array}{c} \frac{2}{7} & \frac{5}{7} & \frac{2}{7} & - \frac{4}{7} \\ \frac{\sqrt{22}}{22} & \frac{\sqrt{22}}{11} & - \frac{2 (2 \sqrt{22})}{2 (11)} & \frac{1}{\sqrt{22}} \end{array}] [\begin{array}{c} 2 & 3 \\ 5 & 7 \\ 2 & - 2 \\ - 4 & - 3 \end{array}]$

Schritt 8.2.2

Kürze den gemeinsamen Faktor.

$[\begin{array}{c} \frac{2}{7} & \frac{5}{7} & \frac{2}{7} & - \frac{4}{7} \\ \frac{\sqrt{22}}{22} & \frac{\sqrt{22}}{11} & - \frac{2 (2 \sqrt{22})}{2 \cdot 11} & \frac{1}{\sqrt{22}} \end{array}] [\begin{array}{c} 2 & 3 \\ 5 & 7 \\ 2 & - 2 \\ - 4 & - 3 \end{array}]$

Schritt 8.2.3

Forme den Ausdruck um.

$[\begin{array}{c} \frac{2}{7} & \frac{5}{7} & \frac{2}{7} & - \frac{4}{7} \\ \frac{\sqrt{22}}{22} & \frac{\sqrt{22}}{11} & - \frac{2 \sqrt{22}}{11} & \frac{1}{\sqrt{22}} \end{array}] [\begin{array}{c} 2 & 3 \\ 5 & 7 \\ 2 & - 2 \\ - 4 & - 3 \end{array}]$

Schritt 9

Mutltipliziere $\frac{1}{\sqrt{22}}$ mit $\frac{\sqrt{22}}{\sqrt{22}}$ .

$[\begin{array}{c} \frac{2}{7} & \frac{5}{7} & \frac{2}{7} & - \frac{4}{7} \\ \frac{\sqrt{22}}{22} & \frac{\sqrt{22}}{11} & - \frac{2 \sqrt{22}}{11} & \frac{1}{\sqrt{22}} \cdot \frac{\sqrt{22}}{\sqrt{22}} \end{array}] [\begin{array}{c} 2 & 3 \\ 5 & 7 \\ 2 & - 2 \\ - 4 & - 3 \end{array}]$

Schritt 10

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Schritt 10.1

Mutltipliziere $\frac{1}{\sqrt{22}}$ mit $\frac{\sqrt{22}}{\sqrt{22}}$ .

$[\begin{array}{c} \frac{2}{7} & \frac{5}{7} & \frac{2}{7} & - \frac{4}{7} \\ \frac{\sqrt{22}}{22} & \frac{\sqrt{22}}{11} & - \frac{2 \sqrt{22}}{11} & \frac{\sqrt{22}}{\sqrt{22} \sqrt{22}} \end{array}] [\begin{array}{c} 2 & 3 \\ 5 & 7 \\ 2 & - 2 \\ - 4 & - 3 \end{array}]$

Schritt 10.2

Potenziere $\sqrt{22}$ mit $1$ .

$[\begin{array}{c} \frac{2}{7} & \frac{5}{7} & \frac{2}{7} & - \frac{4}{7} \\ \frac{\sqrt{22}}{22} & \frac{\sqrt{22}}{11} & - \frac{2 \sqrt{22}}{11} & \frac{\sqrt{22}}{{\sqrt{22}}^{1} \sqrt{22}} \end{array}] [\begin{array}{c} 2 & 3 \\ 5 & 7 \\ 2 & - 2 \\ - 4 & - 3 \end{array}]$

Schritt 10.3

Potenziere $\sqrt{22}$ mit $1$ .

$[\begin{array}{c} \frac{2}{7} & \frac{5}{7} & \frac{2}{7} & - \frac{4}{7} \\ \frac{\sqrt{22}}{22} & \frac{\sqrt{22}}{11} & - \frac{2 \sqrt{22}}{11} & \frac{\sqrt{22}}{{\sqrt{22}}^{1} {\sqrt{22}}^{1}} \end{array}] [\begin{array}{c} 2 & 3 \\ 5 & 7 \\ 2 & - 2 \\ - 4 & - 3 \end{array}]$

Schritt 10.4

Wende die Exponentenregel $a^{m} a^{n} = a^{m + n}$ an, um die Exponenten zu kombinieren.

$[\begin{array}{c} \frac{2}{7} & \frac{5}{7} & \frac{2}{7} & - \frac{4}{7} \\ \frac{\sqrt{22}}{22} & \frac{\sqrt{22}}{11} & - \frac{2 \sqrt{22}}{11} & \frac{\sqrt{22}}{{\sqrt{22}}^{1 + 1}} \end{array}] [\begin{array}{c} 2 & 3 \\ 5 & 7 \\ 2 & - 2 \\ - 4 & - 3 \end{array}]$

Schritt 10.5

Addiere $1$ und $1$ .

$[\begin{array}{c} \frac{2}{7} & \frac{5}{7} & \frac{2}{7} & - \frac{4}{7} \\ \frac{\sqrt{22}}{22} & \frac{\sqrt{22}}{11} & - \frac{2 \sqrt{22}}{11} & \frac{\sqrt{22}}{{\sqrt{22}}^{2}} \end{array}] [\begin{array}{c} 2 & 3 \\ 5 & 7 \\ 2 & - 2 \\ - 4 & - 3 \end{array}]$

Schritt 10.6

Schreibe ${\sqrt{22}}^{2}$ als $22$ um.

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Schritt 10.6.1

Benutze $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ , um $\sqrt{22}$ als $22^{\frac{1}{2}}$ neu zu schreiben.

$[\begin{array}{c} \frac{2}{7} & \frac{5}{7} & \frac{2}{7} & - \frac{4}{7} \\ \frac{\sqrt{22}}{22} & \frac{\sqrt{22}}{11} & - \frac{2 \sqrt{22}}{11} & \frac{\sqrt{22}}{{(22^{\frac{1}{2}})}^{2}} \end{array}] [\begin{array}{c} 2 & 3 \\ 5 & 7 \\ 2 & - 2 \\ - 4 & - 3 \end{array}]$

Schritt 10.6.2

Wende die Potenzregel an und multipliziere die Exponenten, ${(a^{m})}^{n} = a^{m n}$ .

$[\begin{array}{c} \frac{2}{7} & \frac{5}{7} & \frac{2}{7} & - \frac{4}{7} \\ \frac{\sqrt{22}}{22} & \frac{\sqrt{22}}{11} & - \frac{2 \sqrt{22}}{11} & \frac{\sqrt{22}}{22^{\frac{1}{2} \cdot 2}} \end{array}] [\begin{array}{c} 2 & 3 \\ 5 & 7 \\ 2 & - 2 \\ - 4 & - 3 \end{array}]$

Schritt 10.6.3

Kombiniere $\frac{1}{2}$ und $2$ .

$[\begin{array}{c} \frac{2}{7} & \frac{5}{7} & \frac{2}{7} & - \frac{4}{7} \\ \frac{\sqrt{22}}{22} & \frac{\sqrt{22}}{11} & - \frac{2 \sqrt{22}}{11} & \frac{\sqrt{22}}{22^{\frac{2}{2}}} \end{array}] [\begin{array}{c} 2 & 3 \\ 5 & 7 \\ 2 & - 2 \\ - 4 & - 3 \end{array}]$

Schritt 10.6.4

Kürze den gemeinsamen Faktor von $2$ .

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Schritt 10.6.4.1

Kürze den gemeinsamen Faktor.

$[\begin{array}{c} \frac{2}{7} & \frac{5}{7} & \frac{2}{7} & - \frac{4}{7} \\ \frac{\sqrt{22}}{22} & \frac{\sqrt{22}}{11} & - \frac{2 \sqrt{22}}{11} & \frac{\sqrt{22}}{22^{\frac{2}{2}}} \end{array}] [\begin{array}{c} 2 & 3 \\ 5 & 7 \\ 2 & - 2 \\ - 4 & - 3 \end{array}]$

Schritt 10.6.4.2

Forme den Ausdruck um.

$[\begin{array}{c} \frac{2}{7} & \frac{5}{7} & \frac{2}{7} & - \frac{4}{7} \\ \frac{\sqrt{22}}{22} & \frac{\sqrt{22}}{11} & - \frac{2 \sqrt{22}}{11} & \frac{\sqrt{22}}{22^{1}} \end{array}] [\begin{array}{c} 2 & 3 \\ 5 & 7 \\ 2 & - 2 \\ - 4 & - 3 \end{array}]$

Schritt 10.6.5

Berechne den Exponenten.

$[\begin{array}{c} \frac{2}{7} & \frac{5}{7} & \frac{2}{7} & - \frac{4}{7} \\ \frac{\sqrt{22}}{22} & \frac{\sqrt{22}}{11} & - \frac{2 \sqrt{22}}{11} & \frac{\sqrt{22}}{22} \end{array}] [\begin{array}{c} 2 & 3 \\ 5 & 7 \\ 2 & - 2 \\ - 4 & - 3 \end{array}]$

Schritt 11

Multipliziere $[\begin{array}{c} \frac{2}{7} & \frac{5}{7} & \frac{2}{7} & - \frac{4}{7} \\ \frac{\sqrt{22}}{22} & \frac{\sqrt{22}}{11} & - \frac{2 \sqrt{22}}{11} & \frac{\sqrt{22}}{22} \end{array}] [\begin{array}{c} 2 & 3 \\ 5 & 7 \\ 2 & - 2 \\ - 4 & - 3 \end{array}]$ .

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Schritt 11.1

Two matrices can be multiplied if and only if the number of columns in the first matrix is equal to the number of rows in the second matrix. In this case, the first matrix is $2 \times 4$ and the second matrix is $4 \times 2$ .

Schritt 11.2

Multipliziere jede Zeile in der ersten Matrix mit jeder Spalte in der zweiten Matrix.

$[\begin{array}{c} \frac{2}{7} \cdot 2 + \frac{5}{7} \cdot 5 + \frac{2}{7} \cdot 2 - \frac{4}{7} \cdot - 4 & \frac{2}{7} \cdot 3 + \frac{5}{7} \cdot 7 + \frac{2}{7} \cdot - 2 - \frac{4}{7} \cdot - 3 \\ \frac{\sqrt{22}}{22} \cdot 2 + \frac{\sqrt{22}}{11} \cdot 5 - \frac{2 \sqrt{22}}{11} \cdot 2 + \frac{\sqrt{22}}{22} \cdot - 4 & \frac{\sqrt{22}}{22} \cdot 3 + \frac{\sqrt{22}}{11} \cdot 7 - \frac{2 \sqrt{22}}{11} \cdot - 2 + \frac{\sqrt{22}}{22} \cdot - 3 \end{array}]$

Schritt 11.3

Vereinfache jedes Element der Matrix durch Ausmultiplizieren aller Ausdrücke.

$[\begin{array}{c} 7 & 7 \\ 0 & \sqrt{22} \end{array}]$