Lineare Algebra Beispiele

$[\begin{array}{c} - \frac{1}{\sqrt{2}} & - \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{3}} \\ \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{3}} \\ 0 & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{3}} \end{array}] [\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 4 \end{array}] [\begin{array}{c} - \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 \\ - \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 1

Mutltipliziere $\frac{1}{\sqrt{2}}$ mit $\frac{\sqrt{2}}{\sqrt{2}}$ .

$[\begin{array}{c} - (\frac{1}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}) & - \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{3}} \\ \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{3}} \\ 0 & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{3}} \end{array}] [\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 4 \end{array}] [\begin{array}{c} - \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 \\ - \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 2

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Schritt 2.1

Mutltipliziere $\frac{1}{\sqrt{2}}$ mit $\frac{\sqrt{2}}{\sqrt{2}}$ .

$[\begin{array}{c} - \frac{\sqrt{2}}{\sqrt{2} \sqrt{2}} & - \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{3}} \\ \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{3}} \\ 0 & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{3}} \end{array}] [\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 4 \end{array}] [\begin{array}{c} - \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 \\ - \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 2.2

Potenziere $\sqrt{2}$ mit $1$ .

$[\begin{array}{c} - \frac{\sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}} & - \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{3}} \\ \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{3}} \\ 0 & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{3}} \end{array}] [\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 4 \end{array}] [\begin{array}{c} - \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 \\ - \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 2.3

Potenziere $\sqrt{2}$ mit $1$ .

$[\begin{array}{c} - \frac{\sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}} & - \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{3}} \\ \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{3}} \\ 0 & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{3}} \end{array}] [\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 4 \end{array}] [\begin{array}{c} - \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 \\ - \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 2.4

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

$[\begin{array}{c} - \frac{\sqrt{2}}{{\sqrt{2}}^{1 + 1}} & - \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{3}} \\ \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{3}} \\ 0 & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{3}} \end{array}] [\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 4 \end{array}] [\begin{array}{c} - \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 \\ - \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 2.5

Addiere $1$ und $1$ .

$[\begin{array}{c} - \frac{\sqrt{2}}{{\sqrt{2}}^{2}} & - \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{3}} \\ \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{3}} \\ 0 & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{3}} \end{array}] [\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 4 \end{array}] [\begin{array}{c} - \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 \\ - \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 2.6

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

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Schritt 2.6.1

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

$[\begin{array}{c} - \frac{\sqrt{2}}{{(2^{\frac{1}{2}})}^{2}} & - \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{3}} \\ \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{3}} \\ 0 & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{3}} \end{array}] [\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 4 \end{array}] [\begin{array}{c} - \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 \\ - \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 2.6.2

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

$[\begin{array}{c} - \frac{\sqrt{2}}{2^{\frac{1}{2} \cdot 2}} & - \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{3}} \\ \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{3}} \\ 0 & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{3}} \end{array}] [\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 4 \end{array}] [\begin{array}{c} - \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 \\ - \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 2.6.3

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

$[\begin{array}{c} - \frac{\sqrt{2}}{2^{\frac{2}{2}}} & - \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{3}} \\ \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{3}} \\ 0 & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{3}} \end{array}] [\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 4 \end{array}] [\begin{array}{c} - \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 \\ - \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

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.

$[\begin{array}{c} - \frac{\sqrt{2}}{2^{1}} & - \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{3}} \\ \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{3}} \\ 0 & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{3}} \end{array}] [\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 4 \end{array}] [\begin{array}{c} - \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 \\ - \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 2.6.5

Berechne den Exponenten.

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{3}} \\ \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{3}} \\ 0 & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{3}} \end{array}] [\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 4 \end{array}] [\begin{array}{c} - \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 \\ - \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 3

Mutltipliziere $\frac{1}{\sqrt{2}}$ mit $\frac{\sqrt{2}}{\sqrt{2}}$ .

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - (\frac{1}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}) & \frac{1}{\sqrt{3}} \\ \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{3}} \\ 0 & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{3}} \end{array}] [\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 4 \end{array}] [\begin{array}{c} - \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 \\ - \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 4

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Schritt 4.1

Mutltipliziere $\frac{1}{\sqrt{2}}$ mit $\frac{\sqrt{2}}{\sqrt{2}}$ .

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{\sqrt{2} \sqrt{2}} & \frac{1}{\sqrt{3}} \\ \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{3}} \\ 0 & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{3}} \end{array}] [\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 4 \end{array}] [\begin{array}{c} - \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 \\ - \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 4.2

Potenziere $\sqrt{2}$ mit $1$ .

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}} & \frac{1}{\sqrt{3}} \\ \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{3}} \\ 0 & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{3}} \end{array}] [\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 4 \end{array}] [\begin{array}{c} - \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 \\ - \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 4.3

Potenziere $\sqrt{2}$ mit $1$ .

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}} & \frac{1}{\sqrt{3}} \\ \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{3}} \\ 0 & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{3}} \end{array}] [\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 4 \end{array}] [\begin{array}{c} - \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 \\ - \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 4.4

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

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{{\sqrt{2}}^{1 + 1}} & \frac{1}{\sqrt{3}} \\ \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{3}} \\ 0 & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{3}} \end{array}] [\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 4 \end{array}] [\begin{array}{c} - \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 \\ - \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 4.5

Addiere $1$ und $1$ .

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{{\sqrt{2}}^{2}} & \frac{1}{\sqrt{3}} \\ \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{3}} \\ 0 & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{3}} \end{array}] [\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 4 \end{array}] [\begin{array}{c} - \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 \\ - \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 4.6

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

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Schritt 4.6.1

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

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{{(2^{\frac{1}{2}})}^{2}} & \frac{1}{\sqrt{3}} \\ \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{3}} \\ 0 & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{3}} \end{array}] [\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 4 \end{array}] [\begin{array}{c} - \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 \\ - \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 4.6.2

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

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2^{\frac{1}{2} \cdot 2}} & \frac{1}{\sqrt{3}} \\ \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{3}} \\ 0 & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{3}} \end{array}] [\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 4 \end{array}] [\begin{array}{c} - \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 \\ - \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 4.6.3

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

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2^{\frac{2}{2}}} & \frac{1}{\sqrt{3}} \\ \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{3}} \\ 0 & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{3}} \end{array}] [\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 4 \end{array}] [\begin{array}{c} - \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 \\ - \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

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.

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2^{1}} & \frac{1}{\sqrt{3}} \\ \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{3}} \\ 0 & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{3}} \end{array}] [\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 4 \end{array}] [\begin{array}{c} - \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 \\ - \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 4.6.5

Berechne den Exponenten.

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{1}{\sqrt{3}} \\ \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{3}} \\ 0 & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{3}} \end{array}] [\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 4 \end{array}] [\begin{array}{c} - \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 \\ - \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 5

Mutltipliziere $\frac{1}{\sqrt{3}}$ mit $\frac{\sqrt{3}}{\sqrt{3}}$ .

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{1}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} \\ \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{3}} \\ 0 & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{3}} \end{array}] [\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 4 \end{array}] [\begin{array}{c} - \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 \\ - \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 6

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Schritt 6.1

Mutltipliziere $\frac{1}{\sqrt{3}}$ mit $\frac{\sqrt{3}}{\sqrt{3}}$ .

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{\sqrt{3}}{\sqrt{3} \sqrt{3}} \\ \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{3}} \\ 0 & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{3}} \end{array}] [\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 4 \end{array}] [\begin{array}{c} - \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 \\ - \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 6.2

Potenziere $\sqrt{3}$ mit $1$ .

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{\sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}} \\ \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{3}} \\ 0 & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{3}} \end{array}] [\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 4 \end{array}] [\begin{array}{c} - \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 \\ - \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 6.3

Potenziere $\sqrt{3}$ mit $1$ .

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{\sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}} \\ \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{3}} \\ 0 & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{3}} \end{array}] [\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 4 \end{array}] [\begin{array}{c} - \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 \\ - \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 6.4

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

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{\sqrt{3}}{{\sqrt{3}}^{1 + 1}} \\ \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{3}} \\ 0 & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{3}} \end{array}] [\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 4 \end{array}] [\begin{array}{c} - \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 \\ - \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 6.5

Addiere $1$ und $1$ .

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{\sqrt{3}}{{\sqrt{3}}^{2}} \\ \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{3}} \\ 0 & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{3}} \end{array}] [\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 4 \end{array}] [\begin{array}{c} - \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 \\ - \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 6.6

Schreibe ${\sqrt{3}}^{2}$ als $3$ um.

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Schritt 6.6.1

Benutze $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ , um $\sqrt{3}$ als $3^{\frac{1}{2}}$ neu zu schreiben.

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{\sqrt{3}}{{(3^{\frac{1}{2}})}^{2}} \\ \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{3}} \\ 0 & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{3}} \end{array}] [\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 4 \end{array}] [\begin{array}{c} - \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 \\ - \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 6.6.2

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

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{\sqrt{3}}{3^{\frac{1}{2} \cdot 2}} \\ \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{3}} \\ 0 & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{3}} \end{array}] [\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 4 \end{array}] [\begin{array}{c} - \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 \\ - \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 6.6.3

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

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{\sqrt{3}}{3^{\frac{2}{2}}} \\ \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{3}} \\ 0 & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{3}} \end{array}] [\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 4 \end{array}] [\begin{array}{c} - \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 \\ - \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 6.6.4

Kürze den gemeinsamen Faktor von $2$ .

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Schritt 6.6.4.1

Kürze den gemeinsamen Faktor.

Schritt 6.6.4.2

Forme den Ausdruck um.

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{\sqrt{3}}{3^{1}} \\ \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{3}} \\ 0 & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{3}} \end{array}] [\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 4 \end{array}] [\begin{array}{c} - \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 \\ - \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 6.6.5

Berechne den Exponenten.

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{\sqrt{3}}{3} \\ \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{3}} \\ 0 & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{3}} \end{array}] [\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 4 \end{array}] [\begin{array}{c} - \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 \\ - \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 7

Mutltipliziere $\frac{1}{\sqrt{2}}$ mit $\frac{\sqrt{2}}{\sqrt{2}}$ .

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{\sqrt{3}}{3} \\ \frac{1}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} & 0 & \frac{1}{\sqrt{3}} \\ 0 & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{3}} \end{array}] [\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 4 \end{array}] [\begin{array}{c} - \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 \\ - \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 8

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Schritt 8.1

Mutltipliziere $\frac{1}{\sqrt{2}}$ mit $\frac{\sqrt{2}}{\sqrt{2}}$ .

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{\sqrt{3}}{3} \\ \frac{\sqrt{2}}{\sqrt{2} \sqrt{2}} & 0 & \frac{1}{\sqrt{3}} \\ 0 & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{3}} \end{array}] [\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 4 \end{array}] [\begin{array}{c} - \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 \\ - \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 8.2

Potenziere $\sqrt{2}$ mit $1$ .

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{\sqrt{3}}{3} \\ \frac{\sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}} & 0 & \frac{1}{\sqrt{3}} \\ 0 & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{3}} \end{array}] [\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 4 \end{array}] [\begin{array}{c} - \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 \\ - \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 8.3

Potenziere $\sqrt{2}$ mit $1$ .

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{\sqrt{3}}{3} \\ \frac{\sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}} & 0 & \frac{1}{\sqrt{3}} \\ 0 & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{3}} \end{array}] [\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 4 \end{array}] [\begin{array}{c} - \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 \\ - \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 8.4

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

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{\sqrt{3}}{3} \\ \frac{\sqrt{2}}{{\sqrt{2}}^{1 + 1}} & 0 & \frac{1}{\sqrt{3}} \\ 0 & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{3}} \end{array}] [\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 4 \end{array}] [\begin{array}{c} - \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 \\ - \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 8.5

Addiere $1$ und $1$ .

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{\sqrt{3}}{3} \\ \frac{\sqrt{2}}{{\sqrt{2}}^{2}} & 0 & \frac{1}{\sqrt{3}} \\ 0 & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{3}} \end{array}] [\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 4 \end{array}] [\begin{array}{c} - \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 \\ - \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 8.6

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

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Schritt 8.6.1

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

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{\sqrt{3}}{3} \\ \frac{\sqrt{2}}{{(2^{\frac{1}{2}})}^{2}} & 0 & \frac{1}{\sqrt{3}} \\ 0 & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{3}} \end{array}] [\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 4 \end{array}] [\begin{array}{c} - \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 \\ - \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 8.6.2

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

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{\sqrt{3}}{3} \\ \frac{\sqrt{2}}{2^{\frac{1}{2} \cdot 2}} & 0 & \frac{1}{\sqrt{3}} \\ 0 & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{3}} \end{array}] [\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 4 \end{array}] [\begin{array}{c} - \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 \\ - \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 8.6.3

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

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{\sqrt{3}}{3} \\ \frac{\sqrt{2}}{2^{\frac{2}{2}}} & 0 & \frac{1}{\sqrt{3}} \\ 0 & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{3}} \end{array}] [\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 4 \end{array}] [\begin{array}{c} - \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 \\ - \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 8.6.4

Kürze den gemeinsamen Faktor von $2$ .

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Schritt 8.6.4.1

Kürze den gemeinsamen Faktor.

Schritt 8.6.4.2

Forme den Ausdruck um.

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{\sqrt{3}}{3} \\ \frac{\sqrt{2}}{2^{1}} & 0 & \frac{1}{\sqrt{3}} \\ 0 & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{3}} \end{array}] [\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 4 \end{array}] [\begin{array}{c} - \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 \\ - \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 8.6.5

Berechne den Exponenten.

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{\sqrt{3}}{3} \\ \frac{\sqrt{2}}{2} & 0 & \frac{1}{\sqrt{3}} \\ 0 & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{3}} \end{array}] [\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 4 \end{array}] [\begin{array}{c} - \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 \\ - \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 9

Mutltipliziere $\frac{1}{\sqrt{3}}$ mit $\frac{\sqrt{3}}{\sqrt{3}}$ .

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{\sqrt{3}}{3} \\ \frac{\sqrt{2}}{2} & 0 & \frac{1}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} \\ 0 & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{3}} \end{array}] [\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 4 \end{array}] [\begin{array}{c} - \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 \\ - \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 10

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Schritt 10.1

Mutltipliziere $\frac{1}{\sqrt{3}}$ mit $\frac{\sqrt{3}}{\sqrt{3}}$ .

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{\sqrt{3}}{3} \\ \frac{\sqrt{2}}{2} & 0 & \frac{\sqrt{3}}{\sqrt{3} \sqrt{3}} \\ 0 & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{3}} \end{array}] [\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 4 \end{array}] [\begin{array}{c} - \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 \\ - \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 10.2

Potenziere $\sqrt{3}$ mit $1$ .

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{\sqrt{3}}{3} \\ \frac{\sqrt{2}}{2} & 0 & \frac{\sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}} \\ 0 & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{3}} \end{array}] [\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 4 \end{array}] [\begin{array}{c} - \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 \\ - \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 10.3

Potenziere $\sqrt{3}$ mit $1$ .

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{\sqrt{3}}{3} \\ \frac{\sqrt{2}}{2} & 0 & \frac{\sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}} \\ 0 & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{3}} \end{array}] [\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 4 \end{array}] [\begin{array}{c} - \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 \\ - \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 10.4

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

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{\sqrt{3}}{3} \\ \frac{\sqrt{2}}{2} & 0 & \frac{\sqrt{3}}{{\sqrt{3}}^{1 + 1}} \\ 0 & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{3}} \end{array}] [\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 4 \end{array}] [\begin{array}{c} - \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 \\ - \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 10.5

Addiere $1$ und $1$ .

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{\sqrt{3}}{3} \\ \frac{\sqrt{2}}{2} & 0 & \frac{\sqrt{3}}{{\sqrt{3}}^{2}} \\ 0 & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{3}} \end{array}] [\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 4 \end{array}] [\begin{array}{c} - \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 \\ - \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 10.6

Schreibe ${\sqrt{3}}^{2}$ als $3$ um.

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Schritt 10.6.1

Benutze $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ , um $\sqrt{3}$ als $3^{\frac{1}{2}}$ neu zu schreiben.

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{\sqrt{3}}{3} \\ \frac{\sqrt{2}}{2} & 0 & \frac{\sqrt{3}}{{(3^{\frac{1}{2}})}^{2}} \\ 0 & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{3}} \end{array}] [\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 4 \end{array}] [\begin{array}{c} - \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 \\ - \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 10.6.2

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

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{\sqrt{3}}{3} \\ \frac{\sqrt{2}}{2} & 0 & \frac{\sqrt{3}}{3^{\frac{1}{2} \cdot 2}} \\ 0 & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{3}} \end{array}] [\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 4 \end{array}] [\begin{array}{c} - \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 \\ - \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 10.6.3

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

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{\sqrt{3}}{3} \\ \frac{\sqrt{2}}{2} & 0 & \frac{\sqrt{3}}{3^{\frac{2}{2}}} \\ 0 & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{3}} \end{array}] [\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 4 \end{array}] [\begin{array}{c} - \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 \\ - \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 10.6.4

Kürze den gemeinsamen Faktor von $2$ .

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Schritt 10.6.4.1

Kürze den gemeinsamen Faktor.

Schritt 10.6.4.2

Forme den Ausdruck um.

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{\sqrt{3}}{3} \\ \frac{\sqrt{2}}{2} & 0 & \frac{\sqrt{3}}{3^{1}} \\ 0 & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{3}} \end{array}] [\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 4 \end{array}] [\begin{array}{c} - \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 \\ - \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 10.6.5

Berechne den Exponenten.

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{\sqrt{3}}{3} \\ \frac{\sqrt{2}}{2} & 0 & \frac{\sqrt{3}}{3} \\ 0 & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{3}} \end{array}] [\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 4 \end{array}] [\begin{array}{c} - \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 \\ - \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 11

Mutltipliziere $\frac{1}{\sqrt{2}}$ mit $\frac{\sqrt{2}}{\sqrt{2}}$ .

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{\sqrt{3}}{3} \\ \frac{\sqrt{2}}{2} & 0 & \frac{\sqrt{3}}{3} \\ 0 & \frac{1}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} & \frac{1}{\sqrt{3}} \end{array}] [\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 4 \end{array}] [\begin{array}{c} - \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 \\ - \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 12

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Schritt 12.1

Mutltipliziere $\frac{1}{\sqrt{2}}$ mit $\frac{\sqrt{2}}{\sqrt{2}}$ .

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{\sqrt{3}}{3} \\ \frac{\sqrt{2}}{2} & 0 & \frac{\sqrt{3}}{3} \\ 0 & \frac{\sqrt{2}}{\sqrt{2} \sqrt{2}} & \frac{1}{\sqrt{3}} \end{array}] [\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 4 \end{array}] [\begin{array}{c} - \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 \\ - \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 12.2

Potenziere $\sqrt{2}$ mit $1$ .

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{\sqrt{3}}{3} \\ \frac{\sqrt{2}}{2} & 0 & \frac{\sqrt{3}}{3} \\ 0 & \frac{\sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}} & \frac{1}{\sqrt{3}} \end{array}] [\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 4 \end{array}] [\begin{array}{c} - \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 \\ - \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 12.3

Potenziere $\sqrt{2}$ mit $1$ .

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{\sqrt{3}}{3} \\ \frac{\sqrt{2}}{2} & 0 & \frac{\sqrt{3}}{3} \\ 0 & \frac{\sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}} & \frac{1}{\sqrt{3}} \end{array}] [\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 4 \end{array}] [\begin{array}{c} - \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 \\ - \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 12.4

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

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{\sqrt{3}}{3} \\ \frac{\sqrt{2}}{2} & 0 & \frac{\sqrt{3}}{3} \\ 0 & \frac{\sqrt{2}}{{\sqrt{2}}^{1 + 1}} & \frac{1}{\sqrt{3}} \end{array}] [\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 4 \end{array}] [\begin{array}{c} - \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 \\ - \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 12.5

Addiere $1$ und $1$ .

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{\sqrt{3}}{3} \\ \frac{\sqrt{2}}{2} & 0 & \frac{\sqrt{3}}{3} \\ 0 & \frac{\sqrt{2}}{{\sqrt{2}}^{2}} & \frac{1}{\sqrt{3}} \end{array}] [\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 4 \end{array}] [\begin{array}{c} - \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 \\ - \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 12.6

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

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Schritt 12.6.1

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

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{\sqrt{3}}{3} \\ \frac{\sqrt{2}}{2} & 0 & \frac{\sqrt{3}}{3} \\ 0 & \frac{\sqrt{2}}{{(2^{\frac{1}{2}})}^{2}} & \frac{1}{\sqrt{3}} \end{array}] [\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 4 \end{array}] [\begin{array}{c} - \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 \\ - \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 12.6.2

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

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{\sqrt{3}}{3} \\ \frac{\sqrt{2}}{2} & 0 & \frac{\sqrt{3}}{3} \\ 0 & \frac{\sqrt{2}}{2^{\frac{1}{2} \cdot 2}} & \frac{1}{\sqrt{3}} \end{array}] [\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 4 \end{array}] [\begin{array}{c} - \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 \\ - \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 12.6.3

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

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{\sqrt{3}}{3} \\ \frac{\sqrt{2}}{2} & 0 & \frac{\sqrt{3}}{3} \\ 0 & \frac{\sqrt{2}}{2^{\frac{2}{2}}} & \frac{1}{\sqrt{3}} \end{array}] [\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 4 \end{array}] [\begin{array}{c} - \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 \\ - \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 12.6.4

Kürze den gemeinsamen Faktor von $2$ .

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Schritt 12.6.4.1

Kürze den gemeinsamen Faktor.

Schritt 12.6.4.2

Forme den Ausdruck um.

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{\sqrt{3}}{3} \\ \frac{\sqrt{2}}{2} & 0 & \frac{\sqrt{3}}{3} \\ 0 & \frac{\sqrt{2}}{2^{1}} & \frac{1}{\sqrt{3}} \end{array}] [\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 4 \end{array}] [\begin{array}{c} - \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 \\ - \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 12.6.5

Berechne den Exponenten.

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{\sqrt{3}}{3} \\ \frac{\sqrt{2}}{2} & 0 & \frac{\sqrt{3}}{3} \\ 0 & \frac{\sqrt{2}}{2} & \frac{1}{\sqrt{3}} \end{array}] [\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 4 \end{array}] [\begin{array}{c} - \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 \\ - \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 13

Mutltipliziere $\frac{1}{\sqrt{3}}$ mit $\frac{\sqrt{3}}{\sqrt{3}}$ .

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{\sqrt{3}}{3} \\ \frac{\sqrt{2}}{2} & 0 & \frac{\sqrt{3}}{3} \\ 0 & \frac{\sqrt{2}}{2} & \frac{1}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} \end{array}] [\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 4 \end{array}] [\begin{array}{c} - \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 \\ - \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 14

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Schritt 14.1

Mutltipliziere $\frac{1}{\sqrt{3}}$ mit $\frac{\sqrt{3}}{\sqrt{3}}$ .

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{\sqrt{3}}{3} \\ \frac{\sqrt{2}}{2} & 0 & \frac{\sqrt{3}}{3} \\ 0 & \frac{\sqrt{2}}{2} & \frac{\sqrt{3}}{\sqrt{3} \sqrt{3}} \end{array}] [\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 4 \end{array}] [\begin{array}{c} - \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 \\ - \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 14.2

Potenziere $\sqrt{3}$ mit $1$ .

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{\sqrt{3}}{3} \\ \frac{\sqrt{2}}{2} & 0 & \frac{\sqrt{3}}{3} \\ 0 & \frac{\sqrt{2}}{2} & \frac{\sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}} \end{array}] [\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 4 \end{array}] [\begin{array}{c} - \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 \\ - \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 14.3

Potenziere $\sqrt{3}$ mit $1$ .

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{\sqrt{3}}{3} \\ \frac{\sqrt{2}}{2} & 0 & \frac{\sqrt{3}}{3} \\ 0 & \frac{\sqrt{2}}{2} & \frac{\sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}} \end{array}] [\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 4 \end{array}] [\begin{array}{c} - \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 \\ - \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 14.4

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

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{\sqrt{3}}{3} \\ \frac{\sqrt{2}}{2} & 0 & \frac{\sqrt{3}}{3} \\ 0 & \frac{\sqrt{2}}{2} & \frac{\sqrt{3}}{{\sqrt{3}}^{1 + 1}} \end{array}] [\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 4 \end{array}] [\begin{array}{c} - \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 \\ - \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 14.5

Addiere $1$ und $1$ .

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{\sqrt{3}}{3} \\ \frac{\sqrt{2}}{2} & 0 & \frac{\sqrt{3}}{3} \\ 0 & \frac{\sqrt{2}}{2} & \frac{\sqrt{3}}{{\sqrt{3}}^{2}} \end{array}] [\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 4 \end{array}] [\begin{array}{c} - \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 \\ - \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 14.6

Schreibe ${\sqrt{3}}^{2}$ als $3$ um.

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Schritt 14.6.1

Benutze $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ , um $\sqrt{3}$ als $3^{\frac{1}{2}}$ neu zu schreiben.

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{\sqrt{3}}{3} \\ \frac{\sqrt{2}}{2} & 0 & \frac{\sqrt{3}}{3} \\ 0 & \frac{\sqrt{2}}{2} & \frac{\sqrt{3}}{{(3^{\frac{1}{2}})}^{2}} \end{array}] [\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 4 \end{array}] [\begin{array}{c} - \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 \\ - \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 14.6.2

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

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{\sqrt{3}}{3} \\ \frac{\sqrt{2}}{2} & 0 & \frac{\sqrt{3}}{3} \\ 0 & \frac{\sqrt{2}}{2} & \frac{\sqrt{3}}{3^{\frac{1}{2} \cdot 2}} \end{array}] [\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 4 \end{array}] [\begin{array}{c} - \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 \\ - \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 14.6.3

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

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{\sqrt{3}}{3} \\ \frac{\sqrt{2}}{2} & 0 & \frac{\sqrt{3}}{3} \\ 0 & \frac{\sqrt{2}}{2} & \frac{\sqrt{3}}{3^{\frac{2}{2}}} \end{array}] [\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 4 \end{array}] [\begin{array}{c} - \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 \\ - \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 14.6.4

Kürze den gemeinsamen Faktor von $2$ .

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Schritt 14.6.4.1

Kürze den gemeinsamen Faktor.

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{\sqrt{3}}{3} \\ \frac{\sqrt{2}}{2} & 0 & \frac{\sqrt{3}}{3} \\ 0 & \frac{\sqrt{2}}{2} & \frac{\sqrt{3}}{3^{\frac{2}{2}}} \end{array}] [\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 4 \end{array}] [\begin{array}{c} - \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 \\ - \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 14.6.4.2

Forme den Ausdruck um.

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{\sqrt{3}}{3} \\ \frac{\sqrt{2}}{2} & 0 & \frac{\sqrt{3}}{3} \\ 0 & \frac{\sqrt{2}}{2} & \frac{\sqrt{3}}{3^{1}} \end{array}] [\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 4 \end{array}] [\begin{array}{c} - \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 \\ - \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 14.6.5

Berechne den Exponenten.

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{\sqrt{3}}{3} \\ \frac{\sqrt{2}}{2} & 0 & \frac{\sqrt{3}}{3} \\ 0 & \frac{\sqrt{2}}{2} & \frac{\sqrt{3}}{3} \end{array}] [\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 4 \end{array}] [\begin{array}{c} - \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 \\ - \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 15

Multipliziere $[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{\sqrt{3}}{3} \\ \frac{\sqrt{2}}{2} & 0 & \frac{\sqrt{3}}{3} \\ 0 & \frac{\sqrt{2}}{2} & \frac{\sqrt{3}}{3} \end{array}] [\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 4 \end{array}]$ .

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Schritt 15.1

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 $3 \times 3$ and the second matrix is $3 \times 3$ .

Schritt 15.2

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

$[\begin{array}{c} - \frac{\sqrt{2}}{2} \cdot 1 - \frac{\sqrt{2}}{2} \cdot 0 + \frac{\sqrt{3}}{3} \cdot 0 & - \frac{\sqrt{2}}{2} \cdot 0 - \frac{\sqrt{2}}{2} \cdot 1 + \frac{\sqrt{3}}{3} \cdot 0 & - \frac{\sqrt{2}}{2} \cdot 0 - \frac{\sqrt{2}}{2} \cdot 0 + \frac{\sqrt{3}}{3} \cdot 4 \\ \frac{\sqrt{2}}{2} \cdot 1 + 0 \cdot 0 + \frac{\sqrt{3}}{3} \cdot 0 & \frac{\sqrt{2}}{2} \cdot 0 + 0 \cdot 1 + \frac{\sqrt{3}}{3} \cdot 0 & \frac{\sqrt{2}}{2} \cdot 0 + 0 \cdot 0 + \frac{\sqrt{3}}{3} \cdot 4 \\ 0 \cdot 1 + \frac{\sqrt{2}}{2} \cdot 0 + \frac{\sqrt{3}}{3} \cdot 0 & 0 \cdot 0 + \frac{\sqrt{2}}{2} \cdot 1 + \frac{\sqrt{3}}{3} \cdot 0 & 0 \cdot 0 + \frac{\sqrt{2}}{2} \cdot 0 + \frac{\sqrt{3}}{3} \cdot 4 \end{array}] [\begin{array}{c} - \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 \\ - \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 15.3

Vereinfache jedes Element der Matrix durch Ausmultiplizieren aller Ausdrücke.

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \\ \frac{\sqrt{2}}{2} & 0 & \frac{4 \sqrt{3}}{3} \\ 0 & \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \end{array}] [\begin{array}{c} - \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 \\ - \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 16

Mutltipliziere $\frac{1}{\sqrt{2}}$ mit $\frac{\sqrt{2}}{\sqrt{2}}$ .

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \\ \frac{\sqrt{2}}{2} & 0 & \frac{4 \sqrt{3}}{3} \\ 0 & \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \end{array}] [\begin{array}{c} - (\frac{1}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}) & \frac{1}{\sqrt{2}} & 0 \\ - \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 17

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Schritt 17.1

Mutltipliziere $\frac{1}{\sqrt{2}}$ mit $\frac{\sqrt{2}}{\sqrt{2}}$ .

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \\ \frac{\sqrt{2}}{2} & 0 & \frac{4 \sqrt{3}}{3} \\ 0 & \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \end{array}] [\begin{array}{c} - \frac{\sqrt{2}}{\sqrt{2} \sqrt{2}} & \frac{1}{\sqrt{2}} & 0 \\ - \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 17.2

Potenziere $\sqrt{2}$ mit $1$ .

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \\ \frac{\sqrt{2}}{2} & 0 & \frac{4 \sqrt{3}}{3} \\ 0 & \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \end{array}] [\begin{array}{c} - \frac{\sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}} & \frac{1}{\sqrt{2}} & 0 \\ - \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 17.3

Potenziere $\sqrt{2}$ mit $1$ .

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \\ \frac{\sqrt{2}}{2} & 0 & \frac{4 \sqrt{3}}{3} \\ 0 & \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \end{array}] [\begin{array}{c} - \frac{\sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}} & \frac{1}{\sqrt{2}} & 0 \\ - \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 17.4

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

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \\ \frac{\sqrt{2}}{2} & 0 & \frac{4 \sqrt{3}}{3} \\ 0 & \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \end{array}] [\begin{array}{c} - \frac{\sqrt{2}}{{\sqrt{2}}^{1 + 1}} & \frac{1}{\sqrt{2}} & 0 \\ - \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 17.5

Addiere $1$ und $1$ .

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \\ \frac{\sqrt{2}}{2} & 0 & \frac{4 \sqrt{3}}{3} \\ 0 & \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \end{array}] [\begin{array}{c} - \frac{\sqrt{2}}{{\sqrt{2}}^{2}} & \frac{1}{\sqrt{2}} & 0 \\ - \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 17.6

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

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Schritt 17.6.1

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

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \\ \frac{\sqrt{2}}{2} & 0 & \frac{4 \sqrt{3}}{3} \\ 0 & \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \end{array}] [\begin{array}{c} - \frac{\sqrt{2}}{{(2^{\frac{1}{2}})}^{2}} & \frac{1}{\sqrt{2}} & 0 \\ - \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 17.6.2

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

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \\ \frac{\sqrt{2}}{2} & 0 & \frac{4 \sqrt{3}}{3} \\ 0 & \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \end{array}] [\begin{array}{c} - \frac{\sqrt{2}}{2^{\frac{1}{2} \cdot 2}} & \frac{1}{\sqrt{2}} & 0 \\ - \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 17.6.3

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

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \\ \frac{\sqrt{2}}{2} & 0 & \frac{4 \sqrt{3}}{3} \\ 0 & \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \end{array}] [\begin{array}{c} - \frac{\sqrt{2}}{2^{\frac{2}{2}}} & \frac{1}{\sqrt{2}} & 0 \\ - \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 17.6.4

Kürze den gemeinsamen Faktor von $2$ .

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Schritt 17.6.4.1

Kürze den gemeinsamen Faktor.

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \\ \frac{\sqrt{2}}{2} & 0 & \frac{4 \sqrt{3}}{3} \\ 0 & \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \end{array}] [\begin{array}{c} - \frac{\sqrt{2}}{2^{\frac{2}{2}}} & \frac{1}{\sqrt{2}} & 0 \\ - \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 17.6.4.2

Forme den Ausdruck um.

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \\ \frac{\sqrt{2}}{2} & 0 & \frac{4 \sqrt{3}}{3} \\ 0 & \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \end{array}] [\begin{array}{c} - \frac{\sqrt{2}}{2^{1}} & \frac{1}{\sqrt{2}} & 0 \\ - \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 17.6.5

Berechne den Exponenten.

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \\ \frac{\sqrt{2}}{2} & 0 & \frac{4 \sqrt{3}}{3} \\ 0 & \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \end{array}] [\begin{array}{c} - \frac{\sqrt{2}}{2} & \frac{1}{\sqrt{2}} & 0 \\ - \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 18

Mutltipliziere $\frac{1}{\sqrt{2}}$ mit $\frac{\sqrt{2}}{\sqrt{2}}$ .

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \\ \frac{\sqrt{2}}{2} & 0 & \frac{4 \sqrt{3}}{3} \\ 0 & \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \end{array}] [\begin{array}{c} - \frac{\sqrt{2}}{2} & \frac{1}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} & 0 \\ - \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 19

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Schritt 19.1

Mutltipliziere $\frac{1}{\sqrt{2}}$ mit $\frac{\sqrt{2}}{\sqrt{2}}$ .

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \\ \frac{\sqrt{2}}{2} & 0 & \frac{4 \sqrt{3}}{3} \\ 0 & \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \end{array}] [\begin{array}{c} - \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{\sqrt{2} \sqrt{2}} & 0 \\ - \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 19.2

Potenziere $\sqrt{2}$ mit $1$ .

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \\ \frac{\sqrt{2}}{2} & 0 & \frac{4 \sqrt{3}}{3} \\ 0 & \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \end{array}] [\begin{array}{c} - \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}} & 0 \\ - \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 19.3

Potenziere $\sqrt{2}$ mit $1$ .

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \\ \frac{\sqrt{2}}{2} & 0 & \frac{4 \sqrt{3}}{3} \\ 0 & \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \end{array}] [\begin{array}{c} - \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}} & 0 \\ - \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 19.4

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

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \\ \frac{\sqrt{2}}{2} & 0 & \frac{4 \sqrt{3}}{3} \\ 0 & \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \end{array}] [\begin{array}{c} - \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{{\sqrt{2}}^{1 + 1}} & 0 \\ - \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 19.5

Addiere $1$ und $1$ .

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \\ \frac{\sqrt{2}}{2} & 0 & \frac{4 \sqrt{3}}{3} \\ 0 & \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \end{array}] [\begin{array}{c} - \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{{\sqrt{2}}^{2}} & 0 \\ - \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 19.6

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

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Schritt 19.6.1

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

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \\ \frac{\sqrt{2}}{2} & 0 & \frac{4 \sqrt{3}}{3} \\ 0 & \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \end{array}] [\begin{array}{c} - \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{{(2^{\frac{1}{2}})}^{2}} & 0 \\ - \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 19.6.2

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

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \\ \frac{\sqrt{2}}{2} & 0 & \frac{4 \sqrt{3}}{3} \\ 0 & \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \end{array}] [\begin{array}{c} - \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2^{\frac{1}{2} \cdot 2}} & 0 \\ - \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 19.6.3

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

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \\ \frac{\sqrt{2}}{2} & 0 & \frac{4 \sqrt{3}}{3} \\ 0 & \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \end{array}] [\begin{array}{c} - \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2^{\frac{2}{2}}} & 0 \\ - \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 19.6.4

Kürze den gemeinsamen Faktor von $2$ .

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Schritt 19.6.4.1

Kürze den gemeinsamen Faktor.

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \\ \frac{\sqrt{2}}{2} & 0 & \frac{4 \sqrt{3}}{3} \\ 0 & \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \end{array}] [\begin{array}{c} - \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2^{\frac{2}{2}}} & 0 \\ - \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 19.6.4.2

Forme den Ausdruck um.

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \\ \frac{\sqrt{2}}{2} & 0 & \frac{4 \sqrt{3}}{3} \\ 0 & \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \end{array}] [\begin{array}{c} - \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2^{1}} & 0 \\ - \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 19.6.5

Berechne den Exponenten.

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \\ \frac{\sqrt{2}}{2} & 0 & \frac{4 \sqrt{3}}{3} \\ 0 & \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \end{array}] [\begin{array}{c} - \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} & 0 \\ - \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 20

Mutltipliziere $\frac{1}{\sqrt{2}}$ mit $\frac{\sqrt{2}}{\sqrt{2}}$ .

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \\ \frac{\sqrt{2}}{2} & 0 & \frac{4 \sqrt{3}}{3} \\ 0 & \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \end{array}] [\begin{array}{c} - \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} & 0 \\ - (\frac{1}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}) & 0 & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 21

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Schritt 21.1

Mutltipliziere $\frac{1}{\sqrt{2}}$ mit $\frac{\sqrt{2}}{\sqrt{2}}$ .

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \\ \frac{\sqrt{2}}{2} & 0 & \frac{4 \sqrt{3}}{3} \\ 0 & \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \end{array}] [\begin{array}{c} - \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} & 0 \\ - \frac{\sqrt{2}}{\sqrt{2} \sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 21.2

Potenziere $\sqrt{2}$ mit $1$ .

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \\ \frac{\sqrt{2}}{2} & 0 & \frac{4 \sqrt{3}}{3} \\ 0 & \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \end{array}] [\begin{array}{c} - \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} & 0 \\ - \frac{\sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 21.3

Potenziere $\sqrt{2}$ mit $1$ .

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \\ \frac{\sqrt{2}}{2} & 0 & \frac{4 \sqrt{3}}{3} \\ 0 & \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \end{array}] [\begin{array}{c} - \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} & 0 \\ - \frac{\sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}} & 0 & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 21.4

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

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \\ \frac{\sqrt{2}}{2} & 0 & \frac{4 \sqrt{3}}{3} \\ 0 & \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \end{array}] [\begin{array}{c} - \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} & 0 \\ - \frac{\sqrt{2}}{{\sqrt{2}}^{1 + 1}} & 0 & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 21.5

Addiere $1$ und $1$ .

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \\ \frac{\sqrt{2}}{2} & 0 & \frac{4 \sqrt{3}}{3} \\ 0 & \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \end{array}] [\begin{array}{c} - \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} & 0 \\ - \frac{\sqrt{2}}{{\sqrt{2}}^{2}} & 0 & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 21.6

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

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Schritt 21.6.1

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

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \\ \frac{\sqrt{2}}{2} & 0 & \frac{4 \sqrt{3}}{3} \\ 0 & \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \end{array}] [\begin{array}{c} - \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} & 0 \\ - \frac{\sqrt{2}}{{(2^{\frac{1}{2}})}^{2}} & 0 & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 21.6.2

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

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \\ \frac{\sqrt{2}}{2} & 0 & \frac{4 \sqrt{3}}{3} \\ 0 & \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \end{array}] [\begin{array}{c} - \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} & 0 \\ - \frac{\sqrt{2}}{2^{\frac{1}{2} \cdot 2}} & 0 & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 21.6.3

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

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \\ \frac{\sqrt{2}}{2} & 0 & \frac{4 \sqrt{3}}{3} \\ 0 & \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \end{array}] [\begin{array}{c} - \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} & 0 \\ - \frac{\sqrt{2}}{2^{\frac{2}{2}}} & 0 & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 21.6.4

Kürze den gemeinsamen Faktor von $2$ .

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Schritt 21.6.4.1

Kürze den gemeinsamen Faktor.

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \\ \frac{\sqrt{2}}{2} & 0 & \frac{4 \sqrt{3}}{3} \\ 0 & \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \end{array}] [\begin{array}{c} - \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} & 0 \\ - \frac{\sqrt{2}}{2^{\frac{2}{2}}} & 0 & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 21.6.4.2

Forme den Ausdruck um.

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \\ \frac{\sqrt{2}}{2} & 0 & \frac{4 \sqrt{3}}{3} \\ 0 & \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \end{array}] [\begin{array}{c} - \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} & 0 \\ - \frac{\sqrt{2}}{2^{1}} & 0 & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 21.6.5

Berechne den Exponenten.

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \\ \frac{\sqrt{2}}{2} & 0 & \frac{4 \sqrt{3}}{3} \\ 0 & \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \end{array}] [\begin{array}{c} - \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} & 0 \\ - \frac{\sqrt{2}}{2} & 0 & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 22

Mutltipliziere $\frac{1}{\sqrt{2}}$ mit $\frac{\sqrt{2}}{\sqrt{2}}$ .

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \\ \frac{\sqrt{2}}{2} & 0 & \frac{4 \sqrt{3}}{3} \\ 0 & \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \end{array}] [\begin{array}{c} - \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} & 0 \\ - \frac{\sqrt{2}}{2} & 0 & \frac{1}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 23

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Schritt 23.1

Mutltipliziere $\frac{1}{\sqrt{2}}$ mit $\frac{\sqrt{2}}{\sqrt{2}}$ .

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \\ \frac{\sqrt{2}}{2} & 0 & \frac{4 \sqrt{3}}{3} \\ 0 & \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \end{array}] [\begin{array}{c} - \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} & 0 \\ - \frac{\sqrt{2}}{2} & 0 & \frac{\sqrt{2}}{\sqrt{2} \sqrt{2}} \\ \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 23.2

Potenziere $\sqrt{2}$ mit $1$ .

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \\ \frac{\sqrt{2}}{2} & 0 & \frac{4 \sqrt{3}}{3} \\ 0 & \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \end{array}] [\begin{array}{c} - \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} & 0 \\ - \frac{\sqrt{2}}{2} & 0 & \frac{\sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}} \\ \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 23.3

Potenziere $\sqrt{2}$ mit $1$ .

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \\ \frac{\sqrt{2}}{2} & 0 & \frac{4 \sqrt{3}}{3} \\ 0 & \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \end{array}] [\begin{array}{c} - \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} & 0 \\ - \frac{\sqrt{2}}{2} & 0 & \frac{\sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}} \\ \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 23.4

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

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \\ \frac{\sqrt{2}}{2} & 0 & \frac{4 \sqrt{3}}{3} \\ 0 & \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \end{array}] [\begin{array}{c} - \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} & 0 \\ - \frac{\sqrt{2}}{2} & 0 & \frac{\sqrt{2}}{{\sqrt{2}}^{1 + 1}} \\ \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 23.5

Addiere $1$ und $1$ .

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \\ \frac{\sqrt{2}}{2} & 0 & \frac{4 \sqrt{3}}{3} \\ 0 & \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \end{array}] [\begin{array}{c} - \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} & 0 \\ - \frac{\sqrt{2}}{2} & 0 & \frac{\sqrt{2}}{{\sqrt{2}}^{2}} \\ \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 23.6

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

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Schritt 23.6.1

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

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \\ \frac{\sqrt{2}}{2} & 0 & \frac{4 \sqrt{3}}{3} \\ 0 & \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \end{array}] [\begin{array}{c} - \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} & 0 \\ - \frac{\sqrt{2}}{2} & 0 & \frac{\sqrt{2}}{{(2^{\frac{1}{2}})}^{2}} \\ \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 23.6.2

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

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \\ \frac{\sqrt{2}}{2} & 0 & \frac{4 \sqrt{3}}{3} \\ 0 & \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \end{array}] [\begin{array}{c} - \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} & 0 \\ - \frac{\sqrt{2}}{2} & 0 & \frac{\sqrt{2}}{2^{\frac{1}{2} \cdot 2}} \\ \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 23.6.3

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

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \\ \frac{\sqrt{2}}{2} & 0 & \frac{4 \sqrt{3}}{3} \\ 0 & \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \end{array}] [\begin{array}{c} - \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} & 0 \\ - \frac{\sqrt{2}}{2} & 0 & \frac{\sqrt{2}}{2^{\frac{2}{2}}} \\ \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 23.6.4

Kürze den gemeinsamen Faktor von $2$ .

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Schritt 23.6.4.1

Kürze den gemeinsamen Faktor.

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \\ \frac{\sqrt{2}}{2} & 0 & \frac{4 \sqrt{3}}{3} \\ 0 & \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \end{array}] [\begin{array}{c} - \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} & 0 \\ - \frac{\sqrt{2}}{2} & 0 & \frac{\sqrt{2}}{2^{\frac{2}{2}}} \\ \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 23.6.4.2

Forme den Ausdruck um.

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \\ \frac{\sqrt{2}}{2} & 0 & \frac{4 \sqrt{3}}{3} \\ 0 & \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \end{array}] [\begin{array}{c} - \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} & 0 \\ - \frac{\sqrt{2}}{2} & 0 & \frac{\sqrt{2}}{2^{1}} \\ \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 23.6.5

Berechne den Exponenten.

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \\ \frac{\sqrt{2}}{2} & 0 & \frac{4 \sqrt{3}}{3} \\ 0 & \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \end{array}] [\begin{array}{c} - \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} & 0 \\ - \frac{\sqrt{2}}{2} & 0 & \frac{\sqrt{2}}{2} \\ \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 24

Mutltipliziere $\frac{1}{\sqrt{3}}$ mit $\frac{\sqrt{3}}{\sqrt{3}}$ .

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \\ \frac{\sqrt{2}}{2} & 0 & \frac{4 \sqrt{3}}{3} \\ 0 & \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \end{array}] [\begin{array}{c} - \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} & 0 \\ - \frac{\sqrt{2}}{2} & 0 & \frac{\sqrt{2}}{2} \\ \frac{1}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 25

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Schritt 25.1

Mutltipliziere $\frac{1}{\sqrt{3}}$ mit $\frac{\sqrt{3}}{\sqrt{3}}$ .

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \\ \frac{\sqrt{2}}{2} & 0 & \frac{4 \sqrt{3}}{3} \\ 0 & \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \end{array}] [\begin{array}{c} - \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} & 0 \\ - \frac{\sqrt{2}}{2} & 0 & \frac{\sqrt{2}}{2} \\ \frac{\sqrt{3}}{\sqrt{3} \sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 25.2

Potenziere $\sqrt{3}$ mit $1$ .

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \\ \frac{\sqrt{2}}{2} & 0 & \frac{4 \sqrt{3}}{3} \\ 0 & \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \end{array}] [\begin{array}{c} - \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} & 0 \\ - \frac{\sqrt{2}}{2} & 0 & \frac{\sqrt{2}}{2} \\ \frac{\sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 25.3

Potenziere $\sqrt{3}$ mit $1$ .

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \\ \frac{\sqrt{2}}{2} & 0 & \frac{4 \sqrt{3}}{3} \\ 0 & \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \end{array}] [\begin{array}{c} - \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} & 0 \\ - \frac{\sqrt{2}}{2} & 0 & \frac{\sqrt{2}}{2} \\ \frac{\sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 25.4

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

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \\ \frac{\sqrt{2}}{2} & 0 & \frac{4 \sqrt{3}}{3} \\ 0 & \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \end{array}] [\begin{array}{c} - \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} & 0 \\ - \frac{\sqrt{2}}{2} & 0 & \frac{\sqrt{2}}{2} \\ \frac{\sqrt{3}}{{\sqrt{3}}^{1 + 1}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 25.5

Addiere $1$ und $1$ .

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \\ \frac{\sqrt{2}}{2} & 0 & \frac{4 \sqrt{3}}{3} \\ 0 & \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \end{array}] [\begin{array}{c} - \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} & 0 \\ - \frac{\sqrt{2}}{2} & 0 & \frac{\sqrt{2}}{2} \\ \frac{\sqrt{3}}{{\sqrt{3}}^{2}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 25.6

Schreibe ${\sqrt{3}}^{2}$ als $3$ um.

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Schritt 25.6.1

Benutze $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ , um $\sqrt{3}$ als $3^{\frac{1}{2}}$ neu zu schreiben.

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \\ \frac{\sqrt{2}}{2} & 0 & \frac{4 \sqrt{3}}{3} \\ 0 & \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \end{array}] [\begin{array}{c} - \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} & 0 \\ - \frac{\sqrt{2}}{2} & 0 & \frac{\sqrt{2}}{2} \\ \frac{\sqrt{3}}{{(3^{\frac{1}{2}})}^{2}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 25.6.2

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

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \\ \frac{\sqrt{2}}{2} & 0 & \frac{4 \sqrt{3}}{3} \\ 0 & \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \end{array}] [\begin{array}{c} - \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} & 0 \\ - \frac{\sqrt{2}}{2} & 0 & \frac{\sqrt{2}}{2} \\ \frac{\sqrt{3}}{3^{\frac{1}{2} \cdot 2}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 25.6.3

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

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \\ \frac{\sqrt{2}}{2} & 0 & \frac{4 \sqrt{3}}{3} \\ 0 & \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \end{array}] [\begin{array}{c} - \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} & 0 \\ - \frac{\sqrt{2}}{2} & 0 & \frac{\sqrt{2}}{2} \\ \frac{\sqrt{3}}{3^{\frac{2}{2}}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 25.6.4

Kürze den gemeinsamen Faktor von $2$ .

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Schritt 25.6.4.1

Kürze den gemeinsamen Faktor.

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \\ \frac{\sqrt{2}}{2} & 0 & \frac{4 \sqrt{3}}{3} \\ 0 & \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \end{array}] [\begin{array}{c} - \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} & 0 \\ - \frac{\sqrt{2}}{2} & 0 & \frac{\sqrt{2}}{2} \\ \frac{\sqrt{3}}{3^{\frac{2}{2}}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 25.6.4.2

Forme den Ausdruck um.

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \\ \frac{\sqrt{2}}{2} & 0 & \frac{4 \sqrt{3}}{3} \\ 0 & \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \end{array}] [\begin{array}{c} - \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} & 0 \\ - \frac{\sqrt{2}}{2} & 0 & \frac{\sqrt{2}}{2} \\ \frac{\sqrt{3}}{3^{1}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 25.6.5

Berechne den Exponenten.

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \\ \frac{\sqrt{2}}{2} & 0 & \frac{4 \sqrt{3}}{3} \\ 0 & \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \end{array}] [\begin{array}{c} - \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} & 0 \\ - \frac{\sqrt{2}}{2} & 0 & \frac{\sqrt{2}}{2} \\ \frac{\sqrt{3}}{3} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 26

Mutltipliziere $\frac{1}{\sqrt{3}}$ mit $\frac{\sqrt{3}}{\sqrt{3}}$ .

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \\ \frac{\sqrt{2}}{2} & 0 & \frac{4 \sqrt{3}}{3} \\ 0 & \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \end{array}] [\begin{array}{c} - \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} & 0 \\ - \frac{\sqrt{2}}{2} & 0 & \frac{\sqrt{2}}{2} \\ \frac{\sqrt{3}}{3} & \frac{1}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 27

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Schritt 27.1

Mutltipliziere $\frac{1}{\sqrt{3}}$ mit $\frac{\sqrt{3}}{\sqrt{3}}$ .

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \\ \frac{\sqrt{2}}{2} & 0 & \frac{4 \sqrt{3}}{3} \\ 0 & \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \end{array}] [\begin{array}{c} - \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} & 0 \\ - \frac{\sqrt{2}}{2} & 0 & \frac{\sqrt{2}}{2} \\ \frac{\sqrt{3}}{3} & \frac{\sqrt{3}}{\sqrt{3} \sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 27.2

Potenziere $\sqrt{3}$ mit $1$ .

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \\ \frac{\sqrt{2}}{2} & 0 & \frac{4 \sqrt{3}}{3} \\ 0 & \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \end{array}] [\begin{array}{c} - \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} & 0 \\ - \frac{\sqrt{2}}{2} & 0 & \frac{\sqrt{2}}{2} \\ \frac{\sqrt{3}}{3} & \frac{\sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 27.3

Potenziere $\sqrt{3}$ mit $1$ .

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \\ \frac{\sqrt{2}}{2} & 0 & \frac{4 \sqrt{3}}{3} \\ 0 & \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \end{array}] [\begin{array}{c} - \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} & 0 \\ - \frac{\sqrt{2}}{2} & 0 & \frac{\sqrt{2}}{2} \\ \frac{\sqrt{3}}{3} & \frac{\sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 27.4

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

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \\ \frac{\sqrt{2}}{2} & 0 & \frac{4 \sqrt{3}}{3} \\ 0 & \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \end{array}] [\begin{array}{c} - \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} & 0 \\ - \frac{\sqrt{2}}{2} & 0 & \frac{\sqrt{2}}{2} \\ \frac{\sqrt{3}}{3} & \frac{\sqrt{3}}{{\sqrt{3}}^{1 + 1}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 27.5

Addiere $1$ und $1$ .

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \\ \frac{\sqrt{2}}{2} & 0 & \frac{4 \sqrt{3}}{3} \\ 0 & \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \end{array}] [\begin{array}{c} - \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} & 0 \\ - \frac{\sqrt{2}}{2} & 0 & \frac{\sqrt{2}}{2} \\ \frac{\sqrt{3}}{3} & \frac{\sqrt{3}}{{\sqrt{3}}^{2}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 27.6

Schreibe ${\sqrt{3}}^{2}$ als $3$ um.

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Schritt 27.6.1

Benutze $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ , um $\sqrt{3}$ als $3^{\frac{1}{2}}$ neu zu schreiben.

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \\ \frac{\sqrt{2}}{2} & 0 & \frac{4 \sqrt{3}}{3} \\ 0 & \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \end{array}] [\begin{array}{c} - \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} & 0 \\ - \frac{\sqrt{2}}{2} & 0 & \frac{\sqrt{2}}{2} \\ \frac{\sqrt{3}}{3} & \frac{\sqrt{3}}{{(3^{\frac{1}{2}})}^{2}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 27.6.2

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

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \\ \frac{\sqrt{2}}{2} & 0 & \frac{4 \sqrt{3}}{3} \\ 0 & \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \end{array}] [\begin{array}{c} - \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} & 0 \\ - \frac{\sqrt{2}}{2} & 0 & \frac{\sqrt{2}}{2} \\ \frac{\sqrt{3}}{3} & \frac{\sqrt{3}}{3^{\frac{1}{2} \cdot 2}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 27.6.3

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

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \\ \frac{\sqrt{2}}{2} & 0 & \frac{4 \sqrt{3}}{3} \\ 0 & \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \end{array}] [\begin{array}{c} - \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} & 0 \\ - \frac{\sqrt{2}}{2} & 0 & \frac{\sqrt{2}}{2} \\ \frac{\sqrt{3}}{3} & \frac{\sqrt{3}}{3^{\frac{2}{2}}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 27.6.4

Kürze den gemeinsamen Faktor von $2$ .

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Schritt 27.6.4.1

Kürze den gemeinsamen Faktor.

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \\ \frac{\sqrt{2}}{2} & 0 & \frac{4 \sqrt{3}}{3} \\ 0 & \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \end{array}] [\begin{array}{c} - \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} & 0 \\ - \frac{\sqrt{2}}{2} & 0 & \frac{\sqrt{2}}{2} \\ \frac{\sqrt{3}}{3} & \frac{\sqrt{3}}{3^{\frac{2}{2}}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 27.6.4.2

Forme den Ausdruck um.

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \\ \frac{\sqrt{2}}{2} & 0 & \frac{4 \sqrt{3}}{3} \\ 0 & \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \end{array}] [\begin{array}{c} - \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} & 0 \\ - \frac{\sqrt{2}}{2} & 0 & \frac{\sqrt{2}}{2} \\ \frac{\sqrt{3}}{3} & \frac{\sqrt{3}}{3^{1}} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 27.6.5

Berechne den Exponenten.

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \\ \frac{\sqrt{2}}{2} & 0 & \frac{4 \sqrt{3}}{3} \\ 0 & \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \end{array}] [\begin{array}{c} - \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} & 0 \\ - \frac{\sqrt{2}}{2} & 0 & \frac{\sqrt{2}}{2} \\ \frac{\sqrt{3}}{3} & \frac{\sqrt{3}}{3} & \frac{1}{\sqrt{3}} \end{array}]$

Schritt 28

Mutltipliziere $\frac{1}{\sqrt{3}}$ mit $\frac{\sqrt{3}}{\sqrt{3}}$ .

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \\ \frac{\sqrt{2}}{2} & 0 & \frac{4 \sqrt{3}}{3} \\ 0 & \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \end{array}] [\begin{array}{c} - \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} & 0 \\ - \frac{\sqrt{2}}{2} & 0 & \frac{\sqrt{2}}{2} \\ \frac{\sqrt{3}}{3} & \frac{\sqrt{3}}{3} & \frac{1}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} \end{array}]$

Schritt 29

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Schritt 29.1

Mutltipliziere $\frac{1}{\sqrt{3}}$ mit $\frac{\sqrt{3}}{\sqrt{3}}$ .

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \\ \frac{\sqrt{2}}{2} & 0 & \frac{4 \sqrt{3}}{3} \\ 0 & \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \end{array}] [\begin{array}{c} - \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} & 0 \\ - \frac{\sqrt{2}}{2} & 0 & \frac{\sqrt{2}}{2} \\ \frac{\sqrt{3}}{3} & \frac{\sqrt{3}}{3} & \frac{\sqrt{3}}{\sqrt{3} \sqrt{3}} \end{array}]$

Schritt 29.2

Potenziere $\sqrt{3}$ mit $1$ .

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \\ \frac{\sqrt{2}}{2} & 0 & \frac{4 \sqrt{3}}{3} \\ 0 & \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \end{array}] [\begin{array}{c} - \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} & 0 \\ - \frac{\sqrt{2}}{2} & 0 & \frac{\sqrt{2}}{2} \\ \frac{\sqrt{3}}{3} & \frac{\sqrt{3}}{3} & \frac{\sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}} \end{array}]$

Schritt 29.3

Potenziere $\sqrt{3}$ mit $1$ .

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \\ \frac{\sqrt{2}}{2} & 0 & \frac{4 \sqrt{3}}{3} \\ 0 & \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \end{array}] [\begin{array}{c} - \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} & 0 \\ - \frac{\sqrt{2}}{2} & 0 & \frac{\sqrt{2}}{2} \\ \frac{\sqrt{3}}{3} & \frac{\sqrt{3}}{3} & \frac{\sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}} \end{array}]$

Schritt 29.4

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

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \\ \frac{\sqrt{2}}{2} & 0 & \frac{4 \sqrt{3}}{3} \\ 0 & \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \end{array}] [\begin{array}{c} - \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} & 0 \\ - \frac{\sqrt{2}}{2} & 0 & \frac{\sqrt{2}}{2} \\ \frac{\sqrt{3}}{3} & \frac{\sqrt{3}}{3} & \frac{\sqrt{3}}{{\sqrt{3}}^{1 + 1}} \end{array}]$

Schritt 29.5

Addiere $1$ und $1$ .

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \\ \frac{\sqrt{2}}{2} & 0 & \frac{4 \sqrt{3}}{3} \\ 0 & \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \end{array}] [\begin{array}{c} - \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} & 0 \\ - \frac{\sqrt{2}}{2} & 0 & \frac{\sqrt{2}}{2} \\ \frac{\sqrt{3}}{3} & \frac{\sqrt{3}}{3} & \frac{\sqrt{3}}{{\sqrt{3}}^{2}} \end{array}]$

Schritt 29.6

Schreibe ${\sqrt{3}}^{2}$ als $3$ um.

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Schritt 29.6.1

Benutze $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ , um $\sqrt{3}$ als $3^{\frac{1}{2}}$ neu zu schreiben.

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \\ \frac{\sqrt{2}}{2} & 0 & \frac{4 \sqrt{3}}{3} \\ 0 & \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \end{array}] [\begin{array}{c} - \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} & 0 \\ - \frac{\sqrt{2}}{2} & 0 & \frac{\sqrt{2}}{2} \\ \frac{\sqrt{3}}{3} & \frac{\sqrt{3}}{3} & \frac{\sqrt{3}}{{(3^{\frac{1}{2}})}^{2}} \end{array}]$

Schritt 29.6.2

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

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \\ \frac{\sqrt{2}}{2} & 0 & \frac{4 \sqrt{3}}{3} \\ 0 & \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \end{array}] [\begin{array}{c} - \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} & 0 \\ - \frac{\sqrt{2}}{2} & 0 & \frac{\sqrt{2}}{2} \\ \frac{\sqrt{3}}{3} & \frac{\sqrt{3}}{3} & \frac{\sqrt{3}}{3^{\frac{1}{2} \cdot 2}} \end{array}]$

Schritt 29.6.3

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

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \\ \frac{\sqrt{2}}{2} & 0 & \frac{4 \sqrt{3}}{3} \\ 0 & \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \end{array}] [\begin{array}{c} - \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} & 0 \\ - \frac{\sqrt{2}}{2} & 0 & \frac{\sqrt{2}}{2} \\ \frac{\sqrt{3}}{3} & \frac{\sqrt{3}}{3} & \frac{\sqrt{3}}{3^{\frac{2}{2}}} \end{array}]$

Schritt 29.6.4

Kürze den gemeinsamen Faktor von $2$ .

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Schritt 29.6.4.1

Kürze den gemeinsamen Faktor.

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \\ \frac{\sqrt{2}}{2} & 0 & \frac{4 \sqrt{3}}{3} \\ 0 & \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \end{array}] [\begin{array}{c} - \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} & 0 \\ - \frac{\sqrt{2}}{2} & 0 & \frac{\sqrt{2}}{2} \\ \frac{\sqrt{3}}{3} & \frac{\sqrt{3}}{3} & \frac{\sqrt{3}}{3^{\frac{2}{2}}} \end{array}]$

Schritt 29.6.4.2

Forme den Ausdruck um.

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \\ \frac{\sqrt{2}}{2} & 0 & \frac{4 \sqrt{3}}{3} \\ 0 & \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \end{array}] [\begin{array}{c} - \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} & 0 \\ - \frac{\sqrt{2}}{2} & 0 & \frac{\sqrt{2}}{2} \\ \frac{\sqrt{3}}{3} & \frac{\sqrt{3}}{3} & \frac{\sqrt{3}}{3^{1}} \end{array}]$

Schritt 29.6.5

Berechne den Exponenten.

$[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \\ \frac{\sqrt{2}}{2} & 0 & \frac{4 \sqrt{3}}{3} \\ 0 & \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \end{array}] [\begin{array}{c} - \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} & 0 \\ - \frac{\sqrt{2}}{2} & 0 & \frac{\sqrt{2}}{2} \\ \frac{\sqrt{3}}{3} & \frac{\sqrt{3}}{3} & \frac{\sqrt{3}}{3} \end{array}]$

Schritt 30

Multipliziere $[\begin{array}{c} - \frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \\ \frac{\sqrt{2}}{2} & 0 & \frac{4 \sqrt{3}}{3} \\ 0 & \frac{\sqrt{2}}{2} & \frac{4 \sqrt{3}}{3} \end{array}] [\begin{array}{c} - \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} & 0 \\ - \frac{\sqrt{2}}{2} & 0 & \frac{\sqrt{2}}{2} \\ \frac{\sqrt{3}}{3} & \frac{\sqrt{3}}{3} & \frac{\sqrt{3}}{3} \end{array}]$ .

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Schritt 30.1

Schritt 30.2

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

$[\begin{array}{c} - \frac{\sqrt{2}}{2} (- \frac{\sqrt{2}}{2}) - \frac{\sqrt{2}}{2} (- \frac{\sqrt{2}}{2}) + \frac{4 \sqrt{3}}{3} \cdot \frac{\sqrt{3}}{3} & - \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2} \cdot 0 + \frac{4 \sqrt{3}}{3} \cdot \frac{\sqrt{3}}{3} & - \frac{\sqrt{2}}{2} \cdot 0 - \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{2}}{2} + \frac{4 \sqrt{3}}{3} \cdot \frac{\sqrt{3}}{3} \\ \frac{\sqrt{2}}{2} (- \frac{\sqrt{2}}{2}) + 0 (- \frac{\sqrt{2}}{2}) + \frac{4 \sqrt{3}}{3} \cdot \frac{\sqrt{3}}{3} & \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{2}}{2} + 0 \cdot 0 + \frac{4 \sqrt{3}}{3} \cdot \frac{\sqrt{3}}{3} & \frac{\sqrt{2}}{2} \cdot 0 + 0 \frac{\sqrt{2}}{2} + \frac{4 \sqrt{3}}{3} \cdot \frac{\sqrt{3}}{3} \\ 0 (- \frac{\sqrt{2}}{2}) + \frac{\sqrt{2}}{2} (- \frac{\sqrt{2}}{2}) + \frac{4 \sqrt{3}}{3} \cdot \frac{\sqrt{3}}{3} & 0 \frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2} \cdot 0 + \frac{4 \sqrt{3}}{3} \cdot \frac{\sqrt{3}}{3} & 0 \cdot 0 + \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{2}}{2} + \frac{4 \sqrt{3}}{3} \cdot \frac{\sqrt{3}}{3} \end{array}]$

Schritt 30.3

Vereinfache jedes Element der Matrix durch Ausmultiplizieren aller Ausdrücke.

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Schritt 30.3.1

Mutltipliziere $4$ mit $2$ .

$[\begin{array}{c} \frac{7}{3} & \frac{5}{6} & \frac{5}{6} \\ \frac{5}{6} & \frac{11}{6} & \frac{4}{3} \\ \frac{5}{6} & \frac{4}{3} & \frac{3 + 8}{6} \end{array}]$

Schritt 30.3.2

Addiere $3$ und $8$ .

$[\begin{array}{c} \frac{7}{3} & \frac{5}{6} & \frac{5}{6} \\ \frac{5}{6} & \frac{11}{6} & \frac{4}{3} \\ \frac{5}{6} & \frac{4}{3} & \frac{11}{6} \end{array}]$