Elementarmathematik Beispiele

$f (x) = 7 x^{\frac{1}{5}} + 6$ ; find $f^{- 1} (x)$

Schritt 1

Ermittle $f^{- 1} (x)$ .

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Schritt 1.1

Schreibe $f (x) = 7 x^{\frac{1}{5}} + 6$ als Gleichung.

$y = 7 x^{\frac{1}{5}} + 6$

Schritt 1.2

Vertausche die Variablen.

$x = 7 y^{\frac{1}{5}} + 6$

Schritt 1.3

Löse nach $y$ auf.

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Schritt 1.3.1

Schreibe die Gleichung als $7 y^{\frac{1}{5}} + 6 = x$ um.

$7 y^{\frac{1}{5}} + 6 = x$

Schritt 1.3.2

Bringe alle Terme, die nicht $y$ enthalten, auf die rechte Seite der Gleichung.

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Schritt 1.3.2.1

Subtrahiere $6$ von beiden Seiten der Gleichung.

$7 y^{\frac{1}{5}} - x = - 6$

Schritt 1.3.2.2

Addiere $x$ zu beiden Seiten der Gleichung.

$7 y^{\frac{1}{5}} = - 6 + x$

Schritt 1.3.3

Potenziere jede Seite der Gleichung mit $5$ , um den gebrochenen Exponenten auf der linken Seite zu eliminieren.

${(7 y^{\frac{1}{5}})}^{5} = {(- 6 + x)}^{5}$

Schritt 1.3.4

Vereinfache den Exponenten.

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Schritt 1.3.4.1

Vereinfache die linke Seite.

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Schritt 1.3.4.1.1

Vereinfache ${(7 y^{\frac{1}{5}})}^{5}$ .

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Schritt 1.3.4.1.1.1

Wende die Produktregel auf $7 y^{\frac{1}{5}}$ an.

$7^{5} {(y^{\frac{1}{5}})}^{5} = {(- 6 + x)}^{5}$

Schritt 1.3.4.1.1.2

Potenziere $7$ mit $5$ .

$16807 {(y^{\frac{1}{5}})}^{5} = {(- 6 + x)}^{5}$

Schritt 1.3.4.1.1.3

Multipliziere die Exponenten in ${(y^{\frac{1}{5}})}^{5}$ .

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Schritt 1.3.4.1.1.3.1

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

$16807 y^{\frac{1}{5} \cdot 5} = {(- 6 + x)}^{5}$

Schritt 1.3.4.1.1.3.2

Kürze den gemeinsamen Faktor von $5$ .

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Schritt 1.3.4.1.1.3.2.1

Kürze den gemeinsamen Faktor.

$16807 y^{\frac{1}{5} \cdot 5} = {(- 6 + x)}^{5}$

Schritt 1.3.4.1.1.3.2.2

Forme den Ausdruck um.

$16807 y^{1} = {(- 6 + x)}^{5}$

Schritt 1.3.4.1.1.4

Vereinfache.

$16807 y = {(- 6 + x)}^{5}$

Schritt 1.3.4.2

Vereinfache die rechte Seite.

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Schritt 1.3.4.2.1

Vereinfache ${(- 6 + x)}^{5}$ .

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Schritt 1.3.4.2.1.1

Wende den binomischen Lehrsatz an.

$16807 y = {(- 6)}^{5} + 5 {(- 6)}^{4} x + 10 {(- 6)}^{3} x^{2} + 10 {(- 6)}^{2} x^{3} + 5 \cdot - 6 x^{4} + x^{5}$

Schritt 1.3.4.2.1.2

Vereinfache jeden Term.

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Schritt 1.3.4.2.1.2.1

Potenziere $- 6$ mit $5$ .

$16807 y = - 7776 + 5 {(- 6)}^{4} x + 10 {(- 6)}^{3} x^{2} + 10 {(- 6)}^{2} x^{3} + 5 \cdot - 6 x^{4} + x^{5}$

Schritt 1.3.4.2.1.2.2

Potenziere $- 6$ mit $4$ .

$16807 y = - 7776 + 5 \cdot 1296 x + 10 {(- 6)}^{3} x^{2} + 10 {(- 6)}^{2} x^{3} + 5 \cdot - 6 x^{4} + x^{5}$

Schritt 1.3.4.2.1.2.3

Mutltipliziere $5$ mit $1296$ .

$16807 y = - 7776 + 6480 x + 10 {(- 6)}^{3} x^{2} + 10 {(- 6)}^{2} x^{3} + 5 \cdot - 6 x^{4} + x^{5}$

Schritt 1.3.4.2.1.2.4

Potenziere $- 6$ mit $3$ .

$16807 y = - 7776 + 6480 x + 10 \cdot - 216 x^{2} + 10 {(- 6)}^{2} x^{3} + 5 \cdot - 6 x^{4} + x^{5}$

Schritt 1.3.4.2.1.2.5

Mutltipliziere $10$ mit $- 216$ .

$16807 y = - 7776 + 6480 x - 2160 x^{2} + 10 {(- 6)}^{2} x^{3} + 5 \cdot - 6 x^{4} + x^{5}$

Schritt 1.3.4.2.1.2.6

Potenziere $- 6$ mit $2$ .

$16807 y = - 7776 + 6480 x - 2160 x^{2} + 10 \cdot 36 x^{3} + 5 \cdot - 6 x^{4} + x^{5}$

Schritt 1.3.4.2.1.2.7

Mutltipliziere $10$ mit $36$ .

$16807 y = - 7776 + 6480 x - 2160 x^{2} + 360 x^{3} + 5 \cdot - 6 x^{4} + x^{5}$

Schritt 1.3.4.2.1.2.8

Mutltipliziere $5$ mit $- 6$ .

$16807 y = - 7776 + 6480 x - 2160 x^{2} + 360 x^{3} - 30 x^{4} + x^{5}$

Schritt 1.3.5

Teile jeden Ausdruck in $16807 y = - 7776 + 6480 x - 2160 x^{2} + 360 x^{3} - 30 x^{4} + x^{5}$ durch $16807$ und vereinfache.

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Schritt 1.3.5.1

Teile jeden Ausdruck in $16807 y = - 7776 + 6480 x - 2160 x^{2} + 360 x^{3} - 30 x^{4} + x^{5}$ durch $16807$ .

$\frac{16807 y}{16807} = \frac{- 7776}{16807} + \frac{6480 x}{16807} + \frac{- 2160 x^{2}}{16807} + \frac{360 x^{3}}{16807} + \frac{- 30 x^{4}}{16807} + \frac{x^{5}}{16807}$

Schritt 1.3.5.2

Vereinfache die linke Seite.

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Schritt 1.3.5.2.1

Kürze den gemeinsamen Faktor von $16807$ .

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Schritt 1.3.5.2.1.1

Kürze den gemeinsamen Faktor.

$\frac{16807 y}{16807} = \frac{- 7776}{16807} + \frac{6480 x}{16807} + \frac{- 2160 x^{2}}{16807} + \frac{360 x^{3}}{16807} + \frac{- 30 x^{4}}{16807} + \frac{x^{5}}{16807}$

Schritt 1.3.5.2.1.2

Dividiere $y$ durch $1$ .

$y = \frac{- 7776}{16807} + \frac{6480 x}{16807} + \frac{- 2160 x^{2}}{16807} + \frac{360 x^{3}}{16807} + \frac{- 30 x^{4}}{16807} + \frac{x^{5}}{16807}$

Schritt 1.3.5.3

Vereinfache die rechte Seite.

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Schritt 1.3.5.3.1

Vereinfache jeden Term.

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Schritt 1.3.5.3.1.1

Ziehe das Minuszeichen vor den Bruch.

$y = - \frac{7776}{16807} + \frac{6480 x}{16807} + \frac{- 2160 x^{2}}{16807} + \frac{360 x^{3}}{16807} + \frac{- 30 x^{4}}{16807} + \frac{x^{5}}{16807}$

Schritt 1.3.5.3.1.2

Ziehe das Minuszeichen vor den Bruch.

$y = - \frac{7776}{16807} + \frac{6480 x}{16807} - \frac{2160 x^{2}}{16807} + \frac{360 x^{3}}{16807} + \frac{- 30 x^{4}}{16807} + \frac{x^{5}}{16807}$

Schritt 1.3.5.3.1.3

Ziehe das Minuszeichen vor den Bruch.

$y = - \frac{7776}{16807} + \frac{6480 x}{16807} - \frac{2160 x^{2}}{16807} + \frac{360 x^{3}}{16807} - \frac{30 x^{4}}{16807} + \frac{x^{5}}{16807}$

Schritt 1.4

Ersetze $y$ durch $f^{- 1} (x)$ , um die endgültige Lösung anzuzeigen.

$f^{- 1} (x) = - \frac{7776}{16807} + \frac{6480 x}{16807} - \frac{2160 x^{2}}{16807} + \frac{360 x^{3}}{16807} - \frac{30 x^{4}}{16807} + \frac{x^{5}}{16807}$

Schritt 1.5

Überprüfe, ob $f^{- 1} (x) = - \frac{7776}{16807} + \frac{6480 x}{16807} - \frac{2160 x^{2}}{16807} + \frac{360 x^{3}}{16807} - \frac{30 x^{4}}{16807} + \frac{x^{5}}{16807}$ die Umkehrfunktion von $f (x) = 7 x^{\frac{1}{5}} + 6$ ist.

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Schritt 1.5.1

Um die inverse Funktion (Umkehrfunktion) zu prüfen, prüfe ob $f^{- 1} (f (x)) = x$ ist und $f (f^{- 1} (x)) = x$ ist.

Schritt 1.5.2

Berechne $f^{- 1} (f (x))$ .

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Schritt 1.5.2.1

Bilde die verkettete Ergebnisfunktion.

$f^{- 1} (f (x))$

Schritt 1.5.2.2

Berechne $f^{- 1} (7 x^{\frac{1}{5}} + 6)$ durch Einsetzen des Wertes von $f$ in $f^{- 1}$ .

$f^{- 1} (7 x^{\frac{1}{5}} + 6) = - \frac{7776}{16807} + \frac{6480 (7 x^{\frac{1}{5}} + 6)}{16807} - \frac{2160 {(7 x^{\frac{1}{5}} + 6)}^{2}}{16807} + \frac{360 {(7 x^{\frac{1}{5}} + 6)}^{3}}{16807} - \frac{30 {(7 x^{\frac{1}{5}} + 6)}^{4}}{16807} + \frac{{(7 x^{\frac{1}{5}} + 6)}^{5}}{16807}$

Schritt 1.5.2.3

Vereinige die Zähler über dem gemeinsamen Nenner.

$f^{- 1} (7 x^{\frac{1}{5}} + 6) = \frac{- 7776 + 6480 (7 x^{\frac{1}{5}} + 6) - 2160 {(7 x^{\frac{1}{5}} + 6)}^{2} + 360 {(7 x^{\frac{1}{5}} + 6)}^{3} - 30 {(7 x^{\frac{1}{5}} + 6)}^{4} + {(7 x^{\frac{1}{5}} + 6)}^{5}}{16807}$

Schritt 1.5.2.4

Vereinfache jeden Term.

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Schritt 1.5.2.4.1

Wende das Distributivgesetz an.

$f^{- 1} (7 x^{\frac{1}{5}} + 6) = \frac{- 7776 + 6480 (7 x^{\frac{1}{5}}) + 6480 \cdot 6 - 2160 {(7 x^{\frac{1}{5}} + 6)}^{2} + 360 {(7 x^{\frac{1}{5}} + 6)}^{3} - 30 {(7 x^{\frac{1}{5}} + 6)}^{4} + {(7 x^{\frac{1}{5}} + 6)}^{5}}{16807}$

Schritt 1.5.2.4.2

Mutltipliziere $7$ mit $6480$ .

$f^{- 1} (7 x^{\frac{1}{5}} + 6) = \frac{- 7776 + 45360 x^{\frac{1}{5}} + 6480 \cdot 6 - 2160 {(7 x^{\frac{1}{5}} + 6)}^{2} + 360 {(7 x^{\frac{1}{5}} + 6)}^{3} - 30 {(7 x^{\frac{1}{5}} + 6)}^{4} + {(7 x^{\frac{1}{5}} + 6)}^{5}}{16807}$

Schritt 1.5.2.4.3

Mutltipliziere $6480$ mit $6$ .

$f^{- 1} (7 x^{\frac{1}{5}} + 6) = \frac{- 7776 + 45360 x^{\frac{1}{5}} + 38880 - 2160 {(7 x^{\frac{1}{5}} + 6)}^{2} + 360 {(7 x^{\frac{1}{5}} + 6)}^{3} - 30 {(7 x^{\frac{1}{5}} + 6)}^{4} + {(7 x^{\frac{1}{5}} + 6)}^{5}}{16807}$

Schritt 1.5.2.4.4

Schreibe ${(7 x^{\frac{1}{5}} + 6)}^{2}$ als $(7 x^{\frac{1}{5}} + 6) (7 x^{\frac{1}{5}} + 6)$ um.

$f^{- 1} (7 x^{\frac{1}{5}} + 6) = \frac{- 7776 + 45360 x^{\frac{1}{5}} + 38880 - 2160 ((7 x^{\frac{1}{5}} + 6) (7 x^{\frac{1}{5}} + 6)) + 360 {(7 x^{\frac{1}{5}} + 6)}^{3} - 30 {(7 x^{\frac{1}{5}} + 6)}^{4} + {(7 x^{\frac{1}{5}} + 6)}^{5}}{16807}$

Schritt 1.5.2.4.5

Multipliziere $(7 x^{\frac{1}{5}} + 6) (7 x^{\frac{1}{5}} + 6)$ aus unter Verwendung der FOIL-Methode.

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Schritt 1.5.2.4.5.1

Wende das Distributivgesetz an.

$f^{- 1} (7 x^{\frac{1}{5}} + 6) = \frac{- 7776 + 45360 x^{\frac{1}{5}} + 38880 - 2160 (7 x^{\frac{1}{5}} (7 x^{\frac{1}{5}} + 6) + 6 (7 x^{\frac{1}{5}} + 6)) + 360 {(7 x^{\frac{1}{5}} + 6)}^{3} - 30 {(7 x^{\frac{1}{5}} + 6)}^{4} + {(7 x^{\frac{1}{5}} + 6)}^{5}}{16807}$

Schritt 1.5.2.4.5.2

Wende das Distributivgesetz an.

$f^{- 1} (7 x^{\frac{1}{5}} + 6) = \frac{- 7776 + 45360 x^{\frac{1}{5}} + 38880 - 2160 (7 x^{\frac{1}{5}} (7 x^{\frac{1}{5}}) + 7 x^{\frac{1}{5}} \cdot 6 + 6 (7 x^{\frac{1}{5}} + 6)) + 360 {(7 x^{\frac{1}{5}} + 6)}^{3} - 30 {(7 x^{\frac{1}{5}} + 6)}^{4} + {(7 x^{\frac{1}{5}} + 6)}^{5}}{16807}$

Schritt 1.5.2.4.5.3

Wende das Distributivgesetz an.

$f^{- 1} (7 x^{\frac{1}{5}} + 6) = \frac{- 7776 + 45360 x^{\frac{1}{5}} + 38880 - 2160 (7 x^{\frac{1}{5}} (7 x^{\frac{1}{5}}) + 7 x^{\frac{1}{5}} \cdot 6 + 6 (7 x^{\frac{1}{5}}) + 6 \cdot 6) + 360 {(7 x^{\frac{1}{5}} + 6)}^{3} - 30 {(7 x^{\frac{1}{5}} + 6)}^{4} + {(7 x^{\frac{1}{5}} + 6)}^{5}}{16807}$

Schritt 1.5.2.4.6

Vereinfache und fasse gleichartige Terme zusammen.

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Schritt 1.5.2.4.6.1

Vereinfache jeden Term.

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Schritt 1.5.2.4.6.1.1

Schreibe neu unter Anwendung des Kommutativgesetzes der Multiplikation.

$f^{- 1} (7 x^{\frac{1}{5}} + 6) = \frac{- 7776 + 45360 x^{\frac{1}{5}} + 38880 - 2160 (7 \cdot (7 x^{\frac{1}{5}} x^{\frac{1}{5}}) + 7 x^{\frac{1}{5}} \cdot 6 + 6 (7 x^{\frac{1}{5}}) + 6 \cdot 6) + 360 {(7 x^{\frac{1}{5}} + 6)}^{3} - 30 {(7 x^{\frac{1}{5}} + 6)}^{4} + {(7 x^{\frac{1}{5}} + 6)}^{5}}{16807}$

Schritt 1.5.2.4.6.1.2

Multipliziere $x^{\frac{1}{5}}$ mit $x^{\frac{1}{5}}$ durch Addieren der Exponenten.

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Schritt 1.5.2.4.6.1.2.1

Bewege $x^{\frac{1}{5}}$ .

$f^{- 1} (7 x^{\frac{1}{5}} + 6) = \frac{- 7776 + 45360 x^{\frac{1}{5}} + 38880 - 2160 (7 \cdot (7 (x^{\frac{1}{5}} x^{\frac{1}{5}})) + 7 x^{\frac{1}{5}} \cdot 6 + 6 (7 x^{\frac{1}{5}}) + 6 \cdot 6) + 360 {(7 x^{\frac{1}{5}} + 6)}^{3} - 30 {(7 x^{\frac{1}{5}} + 6)}^{4} + {(7 x^{\frac{1}{5}} + 6)}^{5}}{16807}$

Schritt 1.5.2.4.6.1.2.2

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

$f^{- 1} (7 x^{\frac{1}{5}} + 6) = \frac{- 7776 + 45360 x^{\frac{1}{5}} + 38880 - 2160 (7 \cdot (7 x^{\frac{1}{5} + \frac{1}{5}}) + 7 x^{\frac{1}{5}} \cdot 6 + 6 (7 x^{\frac{1}{5}}) + 6 \cdot 6) + 360 {(7 x^{\frac{1}{5}} + 6)}^{3} - 30 {(7 x^{\frac{1}{5}} + 6)}^{4} + {(7 x^{\frac{1}{5}} + 6)}^{5}}{16807}$

Schritt 1.5.2.4.6.1.2.3

Vereinige die Zähler über dem gemeinsamen Nenner.

$f^{- 1} (7 x^{\frac{1}{5}} + 6) = \frac{- 7776 + 45360 x^{\frac{1}{5}} + 38880 - 2160 (7 \cdot (7 x^{\frac{1 + 1}{5}}) + 7 x^{\frac{1}{5}} \cdot 6 + 6 (7 x^{\frac{1}{5}}) + 6 \cdot 6) + 360 {(7 x^{\frac{1}{5}} + 6)}^{3} - 30 {(7 x^{\frac{1}{5}} + 6)}^{4} + {(7 x^{\frac{1}{5}} + 6)}^{5}}{16807}$

Schritt 1.5.2.4.6.1.2.4

Addiere $1$ und $1$ .

$f^{- 1} (7 x^{\frac{1}{5}} + 6) = \frac{- 7776 + 45360 x^{\frac{1}{5}} + 38880 - 2160 (7 \cdot (7 x^{\frac{2}{5}}) + 7 x^{\frac{1}{5}} \cdot 6 + 6 (7 x^{\frac{1}{5}}) + 6 \cdot 6) + 360 {(7 x^{\frac{1}{5}} + 6)}^{3} - 30 {(7 x^{\frac{1}{5}} + 6)}^{4} + {(7 x^{\frac{1}{5}} + 6)}^{5}}{16807}$

Schritt 1.5.2.4.6.1.3

Mutltipliziere $7$ mit $7$ .

$f^{- 1} (7 x^{\frac{1}{5}} + 6) = \frac{- 7776 + 45360 x^{\frac{1}{5}} + 38880 - 2160 (49 x^{\frac{2}{5}} + 7 x^{\frac{1}{5}} \cdot 6 + 6 (7 x^{\frac{1}{5}}) + 6 \cdot 6) + 360 {(7 x^{\frac{1}{5}} + 6)}^{3} - 30 {(7 x^{\frac{1}{5}} + 6)}^{4} + {(7 x^{\frac{1}{5}} + 6)}^{5}}{16807}$

Schritt 1.5.2.4.6.1.4

Mutltipliziere $6$ mit $7$ .

$f^{- 1} (7 x^{\frac{1}{5}} + 6) = \frac{- 7776 + 45360 x^{\frac{1}{5}} + 38880 - 2160 (49 x^{\frac{2}{5}} + 42 x^{\frac{1}{5}} + 6 (7 x^{\frac{1}{5}}) + 6 \cdot 6) + 360 {(7 x^{\frac{1}{5}} + 6)}^{3} - 30 {(7 x^{\frac{1}{5}} + 6)}^{4} + {(7 x^{\frac{1}{5}} + 6)}^{5}}{16807}$

Schritt 1.5.2.4.6.1.5

Mutltipliziere $7$ mit $6$ .

$f^{- 1} (7 x^{\frac{1}{5}} + 6) = \frac{- 7776 + 45360 x^{\frac{1}{5}} + 38880 - 2160 (49 x^{\frac{2}{5}} + 42 x^{\frac{1}{5}} + 42 x^{\frac{1}{5}} + 6 \cdot 6) + 360 {(7 x^{\frac{1}{5}} + 6)}^{3} - 30 {(7 x^{\frac{1}{5}} + 6)}^{4} + {(7 x^{\frac{1}{5}} + 6)}^{5}}{16807}$

Schritt 1.5.2.4.6.1.6

Mutltipliziere $6$ mit $6$ .

$f^{- 1} (7 x^{\frac{1}{5}} + 6) = \frac{- 7776 + 45360 x^{\frac{1}{5}} + 38880 - 2160 (49 x^{\frac{2}{5}} + 42 x^{\frac{1}{5}} + 42 x^{\frac{1}{5}} + 36) + 360 {(7 x^{\frac{1}{5}} + 6)}^{3} - 30 {(7 x^{\frac{1}{5}} + 6)}^{4} + {(7 x^{\frac{1}{5}} + 6)}^{5}}{16807}$

Schritt 1.5.2.4.6.2

Addiere $42 x^{\frac{1}{5}}$ und $42 x^{\frac{1}{5}}$ .

$f^{- 1} (7 x^{\frac{1}{5}} + 6) = \frac{- 7776 + 45360 x^{\frac{1}{5}} + 38880 - 2160 (49 x^{\frac{2}{5}} + 84 x^{\frac{1}{5}} + 36) + 360 {(7 x^{\frac{1}{5}} + 6)}^{3} - 30 {(7 x^{\frac{1}{5}} + 6)}^{4} + {(7 x^{\frac{1}{5}} + 6)}^{5}}{16807}$

Schritt 1.5.2.4.7

Wende das Distributivgesetz an.

$f^{- 1} (7 x^{\frac{1}{5}} + 6) = \frac{- 7776 + 45360 x^{\frac{1}{5}} + 38880 - 2160 (49 x^{\frac{2}{5}}) - 2160 (84 x^{\frac{1}{5}}) - 2160 \cdot 36 + 360 {(7 x^{\frac{1}{5}} + 6)}^{3} - 30 {(7 x^{\frac{1}{5}} + 6)}^{4} + {(7 x^{\frac{1}{5}} + 6)}^{5}}{16807}$

Schritt 1.5.2.4.8

Vereinfache.

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Schritt 1.5.2.4.8.1

Mutltipliziere $49$ mit $- 2160$ .

$f^{- 1} (7 x^{\frac{1}{5}} + 6) = \frac{- 7776 + 45360 x^{\frac{1}{5}} + 38880 - 105840 x^{\frac{2}{5}} - 2160 (84 x^{\frac{1}{5}}) - 2160 \cdot 36 + 360 {(7 x^{\frac{1}{5}} + 6)}^{3} - 30 {(7 x^{\frac{1}{5}} + 6)}^{4} + {(7 x^{\frac{1}{5}} + 6)}^{5}}{16807}$

Schritt 1.5.2.4.8.2

Mutltipliziere $84$ mit $- 2160$ .

$f^{- 1} (7 x^{\frac{1}{5}} + 6) = \frac{- 7776 + 45360 x^{\frac{1}{5}} + 38880 - 105840 x^{\frac{2}{5}} - 181440 x^{\frac{1}{5}} - 2160 \cdot 36 + 360 {(7 x^{\frac{1}{5}} + 6)}^{3} - 30 {(7 x^{\frac{1}{5}} + 6)}^{4} + {(7 x^{\frac{1}{5}} + 6)}^{5}}{16807}$

Schritt 1.5.2.4.8.3

Mutltipliziere $- 2160$ mit $36$ .

$f^{- 1} (7 x^{\frac{1}{5}} + 6) = \frac{- 7776 + 45360 x^{\frac{1}{5}} + 38880 - 105840 x^{\frac{2}{5}} - 181440 x^{\frac{1}{5}} - 77760 + 360 {(7 x^{\frac{1}{5}} + 6)}^{3} - 30 {(7 x^{\frac{1}{5}} + 6)}^{4} + {(7 x^{\frac{1}{5}} + 6)}^{5}}{16807}$

Schritt 1.5.2.4.9

Wende den binomischen Lehrsatz an.

$f^{- 1} (7 x^{\frac{1}{5}} + 6) = \frac{- 7776 + 45360 x^{\frac{1}{5}} + 38880 - 105840 x^{\frac{2}{5}} - 181440 x^{\frac{1}{5}} - 77760 + 360 ({(7 x^{\frac{1}{5}})}^{3} + 3 {(7 x^{\frac{1}{5}})}^{2} \cdot 6 + 3 (7 x^{\frac{1}{5}}) \cdot 6^{2} + 6^{3}) - 30 {(7 x^{\frac{1}{5}} + 6)}^{4} + {(7 x^{\frac{1}{5}} + 6)}^{5}}{16807}$

Schritt 1.5.2.4.10

Vereinfache jeden Term.

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Schritt 1.5.2.4.10.1

Wende die Produktregel auf $7 x^{\frac{1}{5}}$ an.

$f^{- 1} (7 x^{\frac{1}{5}} + 6) = \frac{- 7776 + 45360 x^{\frac{1}{5}} + 38880 - 105840 x^{\frac{2}{5}} - 181440 x^{\frac{1}{5}} - 77760 + 360 (7^{3} {(x^{\frac{1}{5}})}^{3} + 3 {(7 x^{\frac{1}{5}})}^{2} \cdot 6 + 3 (7 x^{\frac{1}{5}}) \cdot 6^{2} + 6^{3}) - 30 {(7 x^{\frac{1}{5}} + 6)}^{4} + {(7 x^{\frac{1}{5}} + 6)}^{5}}{16807}$

Schritt 1.5.2.4.10.2

Potenziere $7$ mit $3$ .

$f^{- 1} (7 x^{\frac{1}{5}} + 6) = \frac{- 7776 + 45360 x^{\frac{1}{5}} + 38880 - 105840 x^{\frac{2}{5}} - 181440 x^{\frac{1}{5}} - 77760 + 360 (343 {(x^{\frac{1}{5}})}^{3} + 3 {(7 x^{\frac{1}{5}})}^{2} \cdot 6 + 3 (7 x^{\frac{1}{5}}) \cdot 6^{2} + 6^{3}) - 30 {(7 x^{\frac{1}{5}} + 6)}^{4} + {(7 x^{\frac{1}{5}} + 6)}^{5}}{16807}$

Schritt 1.5.2.4.10.3

Multipliziere die Exponenten in ${(x^{\frac{1}{5}})}^{3}$ .

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Schritt 1.5.2.4.10.3.1

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

$f^{- 1} (7 x^{\frac{1}{5}} + 6) = \frac{- 7776 + 45360 x^{\frac{1}{5}} + 38880 - 105840 x^{\frac{2}{5}} - 181440 x^{\frac{1}{5}} - 77760 + 360 (343 x^{\frac{1}{5} \cdot 3} + 3 {(7 x^{\frac{1}{5}})}^{2} \cdot 6 + 3 (7 x^{\frac{1}{5}}) \cdot 6^{2} + 6^{3}) - 30 {(7 x^{\frac{1}{5}} + 6)}^{4} + {(7 x^{\frac{1}{5}} + 6)}^{5}}{16807}$

Schritt 1.5.2.4.10.3.2

Kombiniere $\frac{1}{5}$ und $3$ .

$f^{- 1} (7 x^{\frac{1}{5}} + 6) = \frac{- 7776 + 45360 x^{\frac{1}{5}} + 38880 - 105840 x^{\frac{2}{5}} - 181440 x^{\frac{1}{5}} - 77760 + 360 (343 x^{\frac{3}{5}} + 3 {(7 x^{\frac{1}{5}})}^{2} \cdot 6 + 3 (7 x^{\frac{1}{5}}) \cdot 6^{2} + 6^{3}) - 30 {(7 x^{\frac{1}{5}} + 6)}^{4} + {(7 x^{\frac{1}{5}} + 6)}^{5}}{16807}$

Schritt 1.5.2.4.10.4

Wende die Produktregel auf $7 x^{\frac{1}{5}}$ an.

$f^{- 1} (7 x^{\frac{1}{5}} + 6) = \frac{- 7776 + 45360 x^{\frac{1}{5}} + 38880 - 105840 x^{\frac{2}{5}} - 181440 x^{\frac{1}{5}} - 77760 + 360 (343 x^{\frac{3}{5}} + 3 (7^{2} {(x^{\frac{1}{5}})}^{2}) \cdot 6 + 3 (7 x^{\frac{1}{5}}) \cdot 6^{2} + 6^{3}) - 30 {(7 x^{\frac{1}{5}} + 6)}^{4} + {(7 x^{\frac{1}{5}} + 6)}^{5}}{16807}$

Schritt 1.5.2.4.10.5

Potenziere $7$ mit $2$ .

$f^{- 1} (7 x^{\frac{1}{5}} + 6) = \frac{- 7776 + 45360 x^{\frac{1}{5}} + 38880 - 105840 x^{\frac{2}{5}} - 181440 x^{\frac{1}{5}} - 77760 + 360 (343 x^{\frac{3}{5}} + 3 (49 {(x^{\frac{1}{5}})}^{2}) \cdot 6 + 3 (7 x^{\frac{1}{5}}) \cdot 6^{2} + 6^{3}) - 30 {(7 x^{\frac{1}{5}} + 6)}^{4} + {(7 x^{\frac{1}{5}} + 6)}^{5}}{16807}$

Schritt 1.5.2.4.10.6

Multipliziere die Exponenten in ${(x^{\frac{1}{5}})}^{2}$ .

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Schritt 1.5.2.4.10.6.1

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

$f^{- 1} (7 x^{\frac{1}{5}} + 6) = \frac{- 7776 + 45360 x^{\frac{1}{5}} + 38880 - 105840 x^{\frac{2}{5}} - 181440 x^{\frac{1}{5}} - 77760 + 360 (343 x^{\frac{3}{5}} + 3 (49 x^{\frac{1}{5} \cdot 2}) \cdot 6 + 3 (7 x^{\frac{1}{5}}) \cdot 6^{2} + 6^{3}) - 30 {(7 x^{\frac{1}{5}} + 6)}^{4} + {(7 x^{\frac{1}{5}} + 6)}^{5}}{16807}$

Schritt 1.5.2.4.10.6.2

Kombiniere $\frac{1}{5}$ und $2$ .

$f^{- 1} (7 x^{\frac{1}{5}} + 6) = \frac{- 7776 + 45360 x^{\frac{1}{5}} + 38880 - 105840 x^{\frac{2}{5}} - 181440 x^{\frac{1}{5}} - 77760 + 360 (343 x^{\frac{3}{5}} + 3 (49 x^{\frac{2}{5}}) \cdot 6 + 3 (7 x^{\frac{1}{5}}) \cdot 6^{2} + 6^{3}) - 30 {(7 x^{\frac{1}{5}} + 6)}^{4} + {(7 x^{\frac{1}{5}} + 6)}^{5}}{16807}$

Schritt 1.5.2.4.10.7

Mutltipliziere $49$ mit $3$ .

$f^{- 1} (7 x^{\frac{1}{5}} + 6) = \frac{- 7776 + 45360 x^{\frac{1}{5}} + 38880 - 105840 x^{\frac{2}{5}} - 181440 x^{\frac{1}{5}} - 77760 + 360 (343 x^{\frac{3}{5}} + 147 x^{\frac{2}{5}} \cdot 6 + 3 (7 x^{\frac{1}{5}}) \cdot 6^{2} + 6^{3}) - 30 {(7 x^{\frac{1}{5}} + 6)}^{4} + {(7 x^{\frac{1}{5}} + 6)}^{5}}{16807}$

Schritt 1.5.2.4.10.8

Mutltipliziere $6$ mit $147$ .

$f^{- 1} (7 x^{\frac{1}{5}} + 6) = \frac{- 7776 + 45360 x^{\frac{1}{5}} + 38880 - 105840 x^{\frac{2}{5}} - 181440 x^{\frac{1}{5}} - 77760 + 360 (343 x^{\frac{3}{5}} + 882 x^{\frac{2}{5}} + 3 (7 x^{\frac{1}{5}}) \cdot 6^{2} + 6^{3}) - 30 {(7 x^{\frac{1}{5}} + 6)}^{4} + {(7 x^{\frac{1}{5}} + 6)}^{5}}{16807}$

Schritt 1.5.2.4.10.9

Mutltipliziere $7$ mit $3$ .

$f^{- 1} (7 x^{\frac{1}{5}} + 6) = \frac{- 7776 + 45360 x^{\frac{1}{5}} + 38880 - 105840 x^{\frac{2}{5}} - 181440 x^{\frac{1}{5}} - 77760 + 360 (343 x^{\frac{3}{5}} + 882 x^{\frac{2}{5}} + 21 x^{\frac{1}{5}} \cdot 6^{2} + 6^{3}) - 30 {(7 x^{\frac{1}{5}} + 6)}^{4} + {(7 x^{\frac{1}{5}} + 6)}^{5}}{16807}$

Schritt 1.5.2.4.10.10

Potenziere $6$ mit $2$ .

$f^{- 1} (7 x^{\frac{1}{5}} + 6) = \frac{- 7776 + 45360 x^{\frac{1}{5}} + 38880 - 105840 x^{\frac{2}{5}} - 181440 x^{\frac{1}{5}} - 77760 + 360 (343 x^{\frac{3}{5}} + 882 x^{\frac{2}{5}} + 21 x^{\frac{1}{5}} \cdot 36 + 6^{3}) - 30 {(7 x^{\frac{1}{5}} + 6)}^{4} + {(7 x^{\frac{1}{5}} + 6)}^{5}}{16807}$

Schritt 1.5.2.4.10.11

Mutltipliziere $36$ mit $21$ .

$f^{- 1} (7 x^{\frac{1}{5}} + 6) = \frac{- 7776 + 45360 x^{\frac{1}{5}} + 38880 - 105840 x^{\frac{2}{5}} - 181440 x^{\frac{1}{5}} - 77760 + 360 (343 x^{\frac{3}{5}} + 882 x^{\frac{2}{5}} + 756 x^{\frac{1}{5}} + 6^{3}) - 30 {(7 x^{\frac{1}{5}} + 6)}^{4} + {(7 x^{\frac{1}{5}} + 6)}^{5}}{16807}$

Schritt 1.5.2.4.10.12

Potenziere $6$ mit $3$ .

$f^{- 1} (7 x^{\frac{1}{5}} + 6) = \frac{- 7776 + 45360 x^{\frac{1}{5}} + 38880 - 105840 x^{\frac{2}{5}} - 181440 x^{\frac{1}{5}} - 77760 + 360 (343 x^{\frac{3}{5}} + 882 x^{\frac{2}{5}} + 756 x^{\frac{1}{5}} + 216) - 30 {(7 x^{\frac{1}{5}} + 6)}^{4} + {(7 x^{\frac{1}{5}} + 6)}^{5}}{16807}$

Schritt 1.5.2.4.11

Wende das Distributivgesetz an.

$f^{- 1} (7 x^{\frac{1}{5}} + 6) = \frac{- 7776 + 45360 x^{\frac{1}{5}} + 38880 - 105840 x^{\frac{2}{5}} - 181440 x^{\frac{1}{5}} - 77760 + 360 (343 x^{\frac{3}{5}}) + 360 (882 x^{\frac{2}{5}}) + 360 (756 x^{\frac{1}{5}}) + 360 \cdot 216 - 30 {(7 x^{\frac{1}{5}} + 6)}^{4} + {(7 x^{\frac{1}{5}} + 6)}^{5}}{16807}$

Schritt 1.5.2.4.12

Vereinfache.

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Schritt 1.5.2.4.12.1

Mutltipliziere $343$ mit $360$ .

$f^{- 1} (7 x^{\frac{1}{5}} + 6) = \frac{- 7776 + 45360 x^{\frac{1}{5}} + 38880 - 105840 x^{\frac{2}{5}} - 181440 x^{\frac{1}{5}} - 77760 + 123480 x^{\frac{3}{5}} + 360 (882 x^{\frac{2}{5}}) + 360 (756 x^{\frac{1}{5}}) + 360 \cdot 216 - 30 {(7 x^{\frac{1}{5}} + 6)}^{4} + {(7 x^{\frac{1}{5}} + 6)}^{5}}{16807}$

Schritt 1.5.2.4.12.2

Mutltipliziere $882$ mit $360$ .

Schritt 1.5.2.4.12.3

Mutltipliziere $756$ mit $360$ .

Schritt 1.5.2.4.12.4

Mutltipliziere $360$ mit $216$ .

Schritt 1.5.2.4.13

Wende den binomischen Lehrsatz an.

$f^{- 1} (7 x^{\frac{1}{5}} + 6) = \frac{- 7776 + 45360 x^{\frac{1}{5}} + 38880 - 105840 x^{\frac{2}{5}} - 181440 x^{\frac{1}{5}} - 77760 + 123480 x^{\frac{3}{5}} + 317520 x^{\frac{2}{5}} + 272160 x^{\frac{1}{5}} + 77760 - 30 ({(7 x^{\frac{1}{5}})}^{4} + 4 {(7 x^{\frac{1}{5}})}^{3} \cdot 6 + 6 {(7 x^{\frac{1}{5}})}^{2} \cdot 6^{2} + 4 (7 x^{\frac{1}{5}}) \cdot 6^{3} + 6^{4}) + {(7 x^{\frac{1}{5}} + 6)}^{5}}{16807}$

Schritt 1.5.2.4.14

Vereinfache jeden Term.

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Schritt 1.5.2.4.14.1

Wende die Produktregel auf $7 x^{\frac{1}{5}}$ an.

$f^{- 1} (7 x^{\frac{1}{5}} + 6) = \frac{- 7776 + 45360 x^{\frac{1}{5}} + 38880 - 105840 x^{\frac{2}{5}} - 181440 x^{\frac{1}{5}} - 77760 + 123480 x^{\frac{3}{5}} + 317520 x^{\frac{2}{5}} + 272160 x^{\frac{1}{5}} + 77760 - 30 (7^{4} {(x^{\frac{1}{5}})}^{4} + 4 {(7 x^{\frac{1}{5}})}^{3} \cdot 6 + 6 {(7 x^{\frac{1}{5}})}^{2} \cdot 6^{2} + 4 (7 x^{\frac{1}{5}}) \cdot 6^{3} + 6^{4}) + {(7 x^{\frac{1}{5}} + 6)}^{5}}{16807}$

Schritt 1.5.2.4.14.2

Potenziere $7$ mit $4$ .

$f^{- 1} (7 x^{\frac{1}{5}} + 6) = \frac{- 7776 + 45360 x^{\frac{1}{5}} + 38880 - 105840 x^{\frac{2}{5}} - 181440 x^{\frac{1}{5}} - 77760 + 123480 x^{\frac{3}{5}} + 317520 x^{\frac{2}{5}} + 272160 x^{\frac{1}{5}} + 77760 - 30 (2401 {(x^{\frac{1}{5}})}^{4} + 4 {(7 x^{\frac{1}{5}})}^{3} \cdot 6 + 6 {(7 x^{\frac{1}{5}})}^{2} \cdot 6^{2} + 4 (7 x^{\frac{1}{5}}) \cdot 6^{3} + 6^{4}) + {(7 x^{\frac{1}{5}} + 6)}^{5}}{16807}$

Schritt 1.5.2.4.14.3

Multipliziere die Exponenten in ${(x^{\frac{1}{5}})}^{4}$ .

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Schritt 1.5.2.4.14.3.1

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

$f^{- 1} (7 x^{\frac{1}{5}} + 6) = \frac{- 7776 + 45360 x^{\frac{1}{5}} + 38880 - 105840 x^{\frac{2}{5}} - 181440 x^{\frac{1}{5}} - 77760 + 123480 x^{\frac{3}{5}} + 317520 x^{\frac{2}{5}} + 272160 x^{\frac{1}{5}} + 77760 - 30 (2401 x^{\frac{1}{5} \cdot 4} + 4 {(7 x^{\frac{1}{5}})}^{3} \cdot 6 + 6 {(7 x^{\frac{1}{5}})}^{2} \cdot 6^{2} + 4 (7 x^{\frac{1}{5}}) \cdot 6^{3} + 6^{4}) + {(7 x^{\frac{1}{5}} + 6)}^{5}}{16807}$

Schritt 1.5.2.4.14.3.2

Kombiniere $\frac{1}{5}$ und $4$ .

$f^{- 1} (7 x^{\frac{1}{5}} + 6) = \frac{- 7776 + 45360 x^{\frac{1}{5}} + 38880 - 105840 x^{\frac{2}{5}} - 181440 x^{\frac{1}{5}} - 77760 + 123480 x^{\frac{3}{5}} + 317520 x^{\frac{2}{5}} + 272160 x^{\frac{1}{5}} + 77760 - 30 (2401 x^{\frac{4}{5}} + 4 {(7 x^{\frac{1}{5}})}^{3} \cdot 6 + 6 {(7 x^{\frac{1}{5}})}^{2} \cdot 6^{2} + 4 (7 x^{\frac{1}{5}}) \cdot 6^{3} + 6^{4}) + {(7 x^{\frac{1}{5}} + 6)}^{5}}{16807}$

Schritt 1.5.2.4.14.4

Wende die Produktregel auf $7 x^{\frac{1}{5}}$ an.

$f^{- 1} (7 x^{\frac{1}{5}} + 6) = \frac{- 7776 + 45360 x^{\frac{1}{5}} + 38880 - 105840 x^{\frac{2}{5}} - 181440 x^{\frac{1}{5}} - 77760 + 123480 x^{\frac{3}{5}} + 317520 x^{\frac{2}{5}} + 272160 x^{\frac{1}{5}} + 77760 - 30 (2401 x^{\frac{4}{5}} + 4 (7^{3} {(x^{\frac{1}{5}})}^{3}) \cdot 6 + 6 {(7 x^{\frac{1}{5}})}^{2} \cdot 6^{2} + 4 (7 x^{\frac{1}{5}}) \cdot 6^{3} + 6^{4}) + {(7 x^{\frac{1}{5}} + 6)}^{5}}{16807}$

Schritt 1.5.2.4.14.5

Potenziere $7$ mit $3$ .

$f^{- 1} (7 x^{\frac{1}{5}} + 6) = \frac{- 7776 + 45360 x^{\frac{1}{5}} + 38880 - 105840 x^{\frac{2}{5}} - 181440 x^{\frac{1}{5}} - 77760 + 123480 x^{\frac{3}{5}} + 317520 x^{\frac{2}{5}} + 272160 x^{\frac{1}{5}} + 77760 - 30 (2401 x^{\frac{4}{5}} + 4 (343 {(x^{\frac{1}{5}})}^{3}) \cdot 6 + 6 {(7 x^{\frac{1}{5}})}^{2} \cdot 6^{2} + 4 (7 x^{\frac{1}{5}}) \cdot 6^{3} + 6^{4}) + {(7 x^{\frac{1}{5}} + 6)}^{5}}{16807}$

Schritt 1.5.2.4.14.6

Multipliziere die Exponenten in ${(x^{\frac{1}{5}})}^{3}$ .

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Schritt 1.5.2.4.14.6.1

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

$f^{- 1} (7 x^{\frac{1}{5}} + 6) = \frac{- 7776 + 45360 x^{\frac{1}{5}} + 38880 - 105840 x^{\frac{2}{5}} - 181440 x^{\frac{1}{5}} - 77760 + 123480 x^{\frac{3}{5}} + 317520 x^{\frac{2}{5}} + 272160 x^{\frac{1}{5}} + 77760 - 30 (2401 x^{\frac{4}{5}} + 4 (343 x^{\frac{1}{5} \cdot 3}) \cdot 6 + 6 {(7 x^{\frac{1}{5}})}^{2} \cdot 6^{2} + 4 (7 x^{\frac{1}{5}}) \cdot 6^{3} + 6^{4}) + {(7 x^{\frac{1}{5}} + 6)}^{5}}{16807}$

Schritt 1.5.2.4.14.6.2

Kombiniere $\frac{1}{5}$ und $3$ .

$f^{- 1} (7 x^{\frac{1}{5}} + 6) = \frac{- 7776 + 45360 x^{\frac{1}{5}} + 38880 - 105840 x^{\frac{2}{5}} - 181440 x^{\frac{1}{5}} - 77760 + 123480 x^{\frac{3}{5}} + 317520 x^{\frac{2}{5}} + 272160 x^{\frac{1}{5}} + 77760 - 30 (2401 x^{\frac{4}{5}} + 4 (343 x^{\frac{3}{5}}) \cdot 6 + 6 {(7 x^{\frac{1}{5}})}^{2} \cdot 6^{2} + 4 (7 x^{\frac{1}{5}}) \cdot 6^{3} + 6^{4}) + {(7 x^{\frac{1}{5}} + 6)}^{5}}{16807}$

Schritt 1.5.2.4.14.7

Mutltipliziere $343$ mit $4$ .

$f^{- 1} (7 x^{\frac{1}{5}} + 6) = \frac{- 7776 + 45360 x^{\frac{1}{5}} + 38880 - 105840 x^{\frac{2}{5}} - 181440 x^{\frac{1}{5}} - 77760 + 123480 x^{\frac{3}{5}} + 317520 x^{\frac{2}{5}} + 272160 x^{\frac{1}{5}} + 77760 - 30 (2401 x^{\frac{4}{5}} + 1372 x^{\frac{3}{5}} \cdot 6 + 6 {(7 x^{\frac{1}{5}})}^{2} \cdot 6^{2} + 4 (7 x^{\frac{1}{5}}) \cdot 6^{3} + 6^{4}) + {(7 x^{\frac{1}{5}} + 6)}^{5}}{16807}$

Schritt 1.5.2.4.14.8

Mutltipliziere $6$ mit $1372$ .

$f^{- 1} (7 x^{\frac{1}{5}} + 6) = \frac{- 7776 + 45360 x^{\frac{1}{5}} + 38880 - 105840 x^{\frac{2}{5}} - 181440 x^{\frac{1}{5}} - 77760 + 123480 x^{\frac{3}{5}} + 317520 x^{\frac{2}{5}} + 272160 x^{\frac{1}{5}} + 77760 - 30 (2401 x^{\frac{4}{5}} + 8232 x^{\frac{3}{5}} + 6 {(7 x^{\frac{1}{5}})}^{2} \cdot 6^{2} + 4 (7 x^{\frac{1}{5}}) \cdot 6^{3} + 6^{4}) + {(7 x^{\frac{1}{5}} + 6)}^{5}}{16807}$

Schritt 1.5.2.4.14.9

Multipliziere $6$ mit $6^{2}$ durch Addieren der Exponenten.

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Schritt 1.5.2.4.14.9.1

Bewege $6^{2}$ .

$f^{- 1} (7 x^{\frac{1}{5}} + 6) = \frac{- 7776 + 45360 x^{\frac{1}{5}} + 38880 - 105840 x^{\frac{2}{5}} - 181440 x^{\frac{1}{5}} - 77760 + 123480 x^{\frac{3}{5}} + 317520 x^{\frac{2}{5}} + 272160 x^{\frac{1}{5}} + 77760 - 30 (2401 x^{\frac{4}{5}} + 8232 x^{\frac{3}{5}} + 6^{2} \cdot (6 {(7 x^{\frac{1}{5}})}^{2}) + 4 (7 x^{\frac{1}{5}}) \cdot 6^{3} + 6^{4}) + {(7 x^{\frac{1}{5}} + 6)}^{5}}{16807}$

Schritt 1.5.2.4.14.9.2

Mutltipliziere $6^{2}$ mit $6$ .

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Schritt 1.5.2.4.14.9.2.1

Potenziere $6$ mit $1$ .

$f^{- 1} (7 x^{\frac{1}{5}} + 6) = \frac{- 7776 + 45360 x^{\frac{1}{5}} + 38880 - 105840 x^{\frac{2}{5}} - 181440 x^{\frac{1}{5}} - 77760 + 123480 x^{\frac{3}{5}} + 317520 x^{\frac{2}{5}} + 272160 x^{\frac{1}{5}} + 77760 - 30 (2401 x^{\frac{4}{5}} + 8232 x^{\frac{3}{5}} + 6^{2} \cdot (6 {(7 x^{\frac{1}{5}})}^{2}) + 4 (7 x^{\frac{1}{5}}) \cdot 6^{3} + 6^{4}) + {(7 x^{\frac{1}{5}} + 6)}^{5}}{16807}$

Schritt 1.5.2.4.14.9.2.2

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

$f^{- 1} (7 x^{\frac{1}{5}} + 6) = \frac{- 7776 + 45360 x^{\frac{1}{5}} + 38880 - 105840 x^{\frac{2}{5}} - 181440 x^{\frac{1}{5}} - 77760 + 123480 x^{\frac{3}{5}} + 317520 x^{\frac{2}{5}} + 272160 x^{\frac{1}{5}} + 77760 - 30 (2401 x^{\frac{4}{5}} + 8232 x^{\frac{3}{5}} + 6^{2 + 1} {(7 x^{\frac{1}{5}})}^{2} + 4 (7 x^{\frac{1}{5}}) \cdot 6^{3} + 6^{4}) + {(7 x^{\frac{1}{5}} + 6)}^{5}}{16807}$

Schritt 1.5.2.4.14.9.3

Addiere $2$ und $1$ .

$f^{- 1} (7 x^{\frac{1}{5}} + 6) = \frac{- 7776 + 45360 x^{\frac{1}{5}} + 38880 - 105840 x^{\frac{2}{5}} - 181440 x^{\frac{1}{5}} - 77760 + 123480 x^{\frac{3}{5}} + 317520 x^{\frac{2}{5}} + 272160 x^{\frac{1}{5}} + 77760 - 30 (2401 x^{\frac{4}{5}} + 8232 x^{\frac{3}{5}} + 6^{3} {(7 x^{\frac{1}{5}})}^{2} + 4 (7 x^{\frac{1}{5}}) \cdot 6^{3} + 6^{4}) + {(7 x^{\frac{1}{5}} + 6)}^{5}}{16807}$

Schritt 1.5.2.4.14.10

Potenziere $6$ mit $3$ .

$f^{- 1} (7 x^{\frac{1}{5}} + 6) = \frac{- 7776 + 45360 x^{\frac{1}{5}} + 38880 - 105840 x^{\frac{2}{5}} - 181440 x^{\frac{1}{5}} - 77760 + 123480 x^{\frac{3}{5}} + 317520 x^{\frac{2}{5}} + 272160 x^{\frac{1}{5}} + 77760 - 30 (2401 x^{\frac{4}{5}} + 8232 x^{\frac{3}{5}} + 216 {(7 x^{\frac{1}{5}})}^{2} + 4 (7 x^{\frac{1}{5}}) \cdot 6^{3} + 6^{4}) + {(7 x^{\frac{1}{5}} + 6)}^{5}}{16807}$

Schritt 1.5.2.4.14.11

Wende die Produktregel auf $7 x^{\frac{1}{5}}$ an.

$f^{- 1} (7 x^{\frac{1}{5}} + 6) = \frac{- 7776 + 45360 x^{\frac{1}{5}} + 38880 - 105840 x^{\frac{2}{5}} - 181440 x^{\frac{1}{5}} - 77760 + 123480 x^{\frac{3}{5}} + 317520 x^{\frac{2}{5}} + 272160 x^{\frac{1}{5}} + 77760 - 30 (2401 x^{\frac{4}{5}} + 8232 x^{\frac{3}{5}} + 216 (7^{2} {(x^{\frac{1}{5}})}^{2}) + 4 (7 x^{\frac{1}{5}}) \cdot 6^{3} + 6^{4}) + {(7 x^{\frac{1}{5}} + 6)}^{5}}{16807}$

Schritt 1.5.2.4.14.12

Potenziere $7$ mit $2$ .

$f^{- 1} (7 x^{\frac{1}{5}} + 6) = \frac{- 7776 + 45360 x^{\frac{1}{5}} + 38880 - 105840 x^{\frac{2}{5}} - 181440 x^{\frac{1}{5}} - 77760 + 123480 x^{\frac{3}{5}} + 317520 x^{\frac{2}{5}} + 272160 x^{\frac{1}{5}} + 77760 - 30 (2401 x^{\frac{4}{5}} + 8232 x^{\frac{3}{5}} + 216 (49 {(x^{\frac{1}{5}})}^{2}) + 4 (7 x^{\frac{1}{5}}) \cdot 6^{3} + 6^{4}) + {(7 x^{\frac{1}{5}} + 6)}^{5}}{16807}$

Schritt 1.5.2.4.14.13

Multipliziere die Exponenten in ${(x^{\frac{1}{5}})}^{2}$ .

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Schritt 1.5.2.4.14.13.1

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

$f^{- 1} (7 x^{\frac{1}{5}} + 6) = \frac{- 7776 + 45360 x^{\frac{1}{5}} + 38880 - 105840 x^{\frac{2}{5}} - 181440 x^{\frac{1}{5}} - 77760 + 123480 x^{\frac{3}{5}} + 317520 x^{\frac{2}{5}} + 272160 x^{\frac{1}{5}} + 77760 - 30 (2401 x^{\frac{4}{5}} + 8232 x^{\frac{3}{5}} + 216 (49 x^{\frac{1}{5} \cdot 2}) + 4 (7 x^{\frac{1}{5}}) \cdot 6^{3} + 6^{4}) + {(7 x^{\frac{1}{5}} + 6)}^{5}}{16807}$

Schritt 1.5.2.4.14.13.2

Kombiniere $\frac{1}{5}$ und $2$ .

$f^{- 1} (7 x^{\frac{1}{5}} + 6) = \frac{- 7776 + 45360 x^{\frac{1}{5}} + 38880 - 105840 x^{\frac{2}{5}} - 181440 x^{\frac{1}{5}} - 77760 + 123480 x^{\frac{3}{5}} + 317520 x^{\frac{2}{5}} + 272160 x^{\frac{1}{5}} + 77760 - 30 (2401 x^{\frac{4}{5}} + 8232 x^{\frac{3}{5}} + 216 (49 x^{\frac{2}{5}}) + 4 (7 x^{\frac{1}{5}}) \cdot 6^{3} + 6^{4}) + {(7 x^{\frac{1}{5}} + 6)}^{5}}{16807}$

Schritt 1.5.2.4.14.14

Mutltipliziere $49$ mit $216$ .

$f^{- 1} (7 x^{\frac{1}{5}} + 6) = \frac{- 7776 + 45360 x^{\frac{1}{5}} + 38880 - 105840 x^{\frac{2}{5}} - 181440 x^{\frac{1}{5}} - 77760 + 123480 x^{\frac{3}{5}} + 317520 x^{\frac{2}{5}} + 272160 x^{\frac{1}{5}} + 77760 - 30 (2401 x^{\frac{4}{5}} + 8232 x^{\frac{3}{5}} + 10584 x^{\frac{2}{5}} + 4 (7 x^{\frac{1}{5}}) \cdot 6^{3} + 6^{4}) + {(7 x^{\frac{1}{5}} + 6)}^{5}}{16807}$

Schritt 1.5.2.4.14.15

Mutltipliziere $7$ mit $4$ .

$f^{- 1} (7 x^{\frac{1}{5}} + 6) = \frac{- 7776 + 45360 x^{\frac{1}{5}} + 38880 - 105840 x^{\frac{2}{5}} - 181440 x^{\frac{1}{5}} - 77760 + 123480 x^{\frac{3}{5}} + 317520 x^{\frac{2}{5}} + 272160 x^{\frac{1}{5}} + 77760 - 30 (2401 x^{\frac{4}{5}} + 8232 x^{\frac{3}{5}} + 10584 x^{\frac{2}{5}} + 28 x^{\frac{1}{5}} \cdot 6^{3} + 6^{4}) + {(7 x^{\frac{1}{5}} + 6)}^{5}}{16807}$

Schritt 1.5.2.4.14.16

Potenziere $6$ mit $3$ .

Schritt 1.5.2.4.14.17

Mutltipliziere $216$ mit $28$ .

Schritt 1.5.2.4.14.18

Potenziere $6$ mit $4$ .

Schritt 1.5.2.4.15

Wende das Distributivgesetz an.

$f^{- 1} (7 x^{\frac{1}{5}} + 6) = \frac{- 7776 + 45360 x^{\frac{1}{5}} + 38880 - 105840 x^{\frac{2}{5}} - 181440 x^{\frac{1}{5}} - 77760 + 123480 x^{\frac{3}{5}} + 317520 x^{\frac{2}{5}} + 272160 x^{\frac{1}{5}} + 77760 - 30 (2401 x^{\frac{4}{5}}) - 30 (8232 x^{\frac{3}{5}}) - 30 (10584 x^{\frac{2}{5}}) - 30 (6048 x^{\frac{1}{5}}) - 30 \cdot 1296 + {(7 x^{\frac{1}{5}} + 6)}^{5}}{16807}$

Schritt 1.5.2.4.16

Vereinfache.

Tippen, um mehr Schritte zu sehen ...

Schritt 1.5.2.4.16.1

Mutltipliziere $2401$ mit $- 30$ .

$f^{- 1} (7 x^{\frac{1}{5}} + 6) = \frac{- 7776 + 45360 x^{\frac{1}{5}} + 38880 - 105840 x^{\frac{2}{5}} - 181440 x^{\frac{1}{5}} - 77760 + 123480 x^{\frac{3}{5}} + 317520 x^{\frac{2}{5}} + 272160 x^{\frac{1}{5}} + 77760 - 72030 x^{\frac{4}{5}} - 30 (8232 x^{\frac{3}{5}}) - 30 (10584 x^{\frac{2}{5}}) - 30 (6048 x^{\frac{1}{5}}) - 30 \cdot 1296 + {(7 x^{\frac{1}{5}} + 6)}^{5}}{16807}$

Schritt 1.5.2.4.16.2

Mutltipliziere $8232$ mit $- 30$ .

$f^{- 1} (7 x^{\frac{1}{5}} + 6) = \frac{- 7776 + 45360 x^{\frac{1}{5}} + 38880 - 105840 x^{\frac{2}{5}} - 181440 x^{\frac{1}{5}} - 77760 + 123480 x^{\frac{3}{5}} + 317520 x^{\frac{2}{5}} + 272160 x^{\frac{1}{5}} + 77760 - 72030 x^{\frac{4}{5}} - 246960 x^{\frac{3}{5}} - 30 (10584 x^{\frac{2}{5}}) - 30 (6048 x^{\frac{1}{5}}) - 30 \cdot 1296 + {(7 x^{\frac{1}{5}} + 6)}^{5}}{16807}$

Schritt 1.5.2.4.16.3

Mutltipliziere $10584$ mit $- 30$ .

$f^{- 1} (7 x^{\frac{1}{5}} + 6) = \frac{- 7776 + 45360 x^{\frac{1}{5}} + 38880 - 105840 x^{\frac{2}{5}} - 181440 x^{\frac{1}{5}} - 77760 + 123480 x^{\frac{3}{5}} + 317520 x^{\frac{2}{5}} + 272160 x^{\frac{1}{5}} + 77760 - 72030 x^{\frac{4}{5}} - 246960 x^{\frac{3}{5}} - 317520 x^{\frac{2}{5}} - 30 (6048 x^{\frac{1}{5}}) - 30 \cdot 1296 + {(7 x^{\frac{1}{5}} + 6)}^{5}}{16807}$

Schritt 1.5.2.4.16.4

Mutltipliziere $6048$ mit $- 30$ .

$f^{- 1} (7 x^{\frac{1}{5}} + 6) = \frac{- 7776 + 45360 x^{\frac{1}{5}} + 38880 - 105840 x^{\frac{2}{5}} - 181440 x^{\frac{1}{5}} - 77760 + 123480 x^{\frac{3}{5}} + 317520 x^{\frac{2}{5}} + 272160 x^{\frac{1}{5}} + 77760 - 72030 x^{\frac{4}{5}} - 246960 x^{\frac{3}{5}} - 317520 x^{\frac{2}{5}} - 181440 x^{\frac{1}{5}} - 30 \cdot 1296 + {(7 x^{\frac{1}{5}} + 6)}^{5}}{16807}$

Schritt 1.5.2.4.16.5

Mutltipliziere $- 30$ mit $1296$ .

Schritt 1.5.2.4.17

Wende den binomischen Lehrsatz an.

$f^{- 1} (7 x^{\frac{1}{5}} + 6) = \frac{- 7776 + 45360 x^{\frac{1}{5}} + 38880 - 105840 x^{\frac{2}{5}} - 181440 x^{\frac{1}{5}} - 77760 + 123480 x^{\frac{3}{5}} + 317520 x^{\frac{2}{5}} + 272160 x^{\frac{1}{5}} + 77760 - 72030 x^{\frac{4}{5}} - 246960 x^{\frac{3}{5}} - 317520 x^{\frac{2}{5}} - 181440 x^{\frac{1}{5}} - 38880 + {(7 x^{\frac{1}{5}})}^{5} + 5 {(7 x^{\frac{1}{5}})}^{4} \cdot 6 + 10 {(7 x^{\frac{1}{5}})}^{3} \cdot 6^{2} + 10 {(7 x^{\frac{1}{5}})}^{2} \cdot 6^{3} + 5 (7 x^{\frac{1}{5}}) \cdot 6^{4} + 6^{5}}{16807}$

Schritt 1.5.2.4.18

Vereinfache jeden Term.

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Schritt 1.5.2.4.18.1

Wende die Produktregel auf $7 x^{\frac{1}{5}}$ an.

$f^{- 1} (7 x^{\frac{1}{5}} + 6) = \frac{- 7776 + 45360 x^{\frac{1}{5}} + 38880 - 105840 x^{\frac{2}{5}} - 181440 x^{\frac{1}{5}} - 77760 + 123480 x^{\frac{3}{5}} + 317520 x^{\frac{2}{5}} + 272160 x^{\frac{1}{5}} + 77760 - 72030 x^{\frac{4}{5}} - 246960 x^{\frac{3}{5}} - 317520 x^{\frac{2}{5}} - 181440 x^{\frac{1}{5}} - 38880 + 7^{5} {(x^{\frac{1}{5}})}^{5} + 5 {(7 x^{\frac{1}{5}})}^{4} \cdot 6 + 10 {(7 x^{\frac{1}{5}})}^{3} \cdot 6^{2} + 10 {(7 x^{\frac{1}{5}})}^{2} \cdot 6^{3} + 5 (7 x^{\frac{1}{5}}) \cdot 6^{4} + 6^{5}}{16807}$

Schritt 1.5.2.4.18.2

Potenziere $7$ mit $5$ .

$f^{- 1} (7 x^{\frac{1}{5}} + 6) = \frac{- 7776 + 45360 x^{\frac{1}{5}} + 38880 - 105840 x^{\frac{2}{5}} - 181440 x^{\frac{1}{5}} - 77760 + 123480 x^{\frac{3}{5}} + 317520 x^{\frac{2}{5}} + 272160 x^{\frac{1}{5}} + 77760 - 72030 x^{\frac{4}{5}} - 246960 x^{\frac{3}{5}} - 317520 x^{\frac{2}{5}} - 181440 x^{\frac{1}{5}} - 38880 + 16807 {(x^{\frac{1}{5}})}^{5} + 5 {(7 x^{\frac{1}{5}})}^{4} \cdot 6 + 10 {(7 x^{\frac{1}{5}})}^{3} \cdot 6^{2} + 10 {(7 x^{\frac{1}{5}})}^{2} \cdot 6^{3} + 5 (7 x^{\frac{1}{5}}) \cdot 6^{4} + 6^{5}}{16807}$

Schritt 1.5.2.4.18.3

Multipliziere die Exponenten in ${(x^{\frac{1}{5}})}^{5}$ .

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Schritt 1.5.2.4.18.3.1

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

$f^{- 1} (7 x^{\frac{1}{5}} + 6) = \frac{- 7776 + 45360 x^{\frac{1}{5}} + 38880 - 105840 x^{\frac{2}{5}} - 181440 x^{\frac{1}{5}} - 77760 + 123480 x^{\frac{3}{5}} + 317520 x^{\frac{2}{5}} + 272160 x^{\frac{1}{5}} + 77760 - 72030 x^{\frac{4}{5}} - 246960 x^{\frac{3}{5}} - 317520 x^{\frac{2}{5}} - 181440 x^{\frac{1}{5}} - 38880 + 16807 x^{\frac{1}{5} \cdot 5} + 5 {(7 x^{\frac{1}{5}})}^{4} \cdot 6 + 10 {(7 x^{\frac{1}{5}})}^{3} \cdot 6^{2} + 10 {(7 x^{\frac{1}{5}})}^{2} \cdot 6^{3} + 5 (7 x^{\frac{1}{5}}) \cdot 6^{4} + 6^{5}}{16807}$

Schritt 1.5.2.4.18.3.2

Kürze den gemeinsamen Faktor von $5$ .

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Schritt 1.5.2.4.18.3.2.1

Kürze den gemeinsamen Faktor.

$f^{- 1} (7 x^{\frac{1}{5}} + 6) = \frac{- 7776 + 45360 x^{\frac{1}{5}} + 38880 - 105840 x^{\frac{2}{5}} - 181440 x^{\frac{1}{5}} - 77760 + 123480 x^{\frac{3}{5}} + 317520 x^{\frac{2}{5}} + 272160 x^{\frac{1}{5}} + 77760 - 72030 x^{\frac{4}{5}} - 246960 x^{\frac{3}{5}} - 317520 x^{\frac{2}{5}} - 181440 x^{\frac{1}{5}} - 38880 + 16807 x^{\frac{1}{5} \cdot 5} + 5 {(7 x^{\frac{1}{5}})}^{4} \cdot 6 + 10 {(7 x^{\frac{1}{5}})}^{3} \cdot 6^{2} + 10 {(7 x^{\frac{1}{5}})}^{2} \cdot 6^{3} + 5 (7 x^{\frac{1}{5}}) \cdot 6^{4} + 6^{5}}{16807}$

Schritt 1.5.2.4.18.3.2.2

Forme den Ausdruck um.

$f^{- 1} (7 x^{\frac{1}{5}} + 6) = \frac{- 7776 + 45360 x^{\frac{1}{5}} + 38880 - 105840 x^{\frac{2}{5}} - 181440 x^{\frac{1}{5}} - 77760 + 123480 x^{\frac{3}{5}} + 317520 x^{\frac{2}{5}} + 272160 x^{\frac{1}{5}} + 77760 - 72030 x^{\frac{4}{5}} - 246960 x^{\frac{3}{5}} - 317520 x^{\frac{2}{5}} - 181440 x^{\frac{1}{5}} - 38880 + 16807 x + 5 {(7 x^{\frac{1}{5}})}^{4} \cdot 6 + 10 {(7 x^{\frac{1}{5}})}^{3} \cdot 6^{2} + 10 {(7 x^{\frac{1}{5}})}^{2} \cdot 6^{3} + 5 (7 x^{\frac{1}{5}}) \cdot 6^{4} + 6^{5}}{16807}$

Schritt 1.5.2.4.18.4

Vereinfache.

$f^{- 1} (7 x^{\frac{1}{5}} + 6) = \frac{- 7776 + 45360 x^{\frac{1}{5}} + 38880 - 105840 x^{\frac{2}{5}} - 181440 x^{\frac{1}{5}} - 77760 + 123480 x^{\frac{3}{5}} + 317520 x^{\frac{2}{5}} + 272160 x^{\frac{1}{5}} + 77760 - 72030 x^{\frac{4}{5}} - 246960 x^{\frac{3}{5}} - 317520 x^{\frac{2}{5}} - 181440 x^{\frac{1}{5}} - 38880 + 16807 x + 5 {(7 x^{\frac{1}{5}})}^{4} \cdot 6 + 10 {(7 x^{\frac{1}{5}})}^{3} \cdot 6^{2} + 10 {(7 x^{\frac{1}{5}})}^{2} \cdot 6^{3} + 5 (7 x^{\frac{1}{5}}) \cdot 6^{4} + 6^{5}}{16807}$

Schritt 1.5.2.4.18.5

Wende die Produktregel auf $7 x^{\frac{1}{5}}$ an.

$f^{- 1} (7 x^{\frac{1}{5}} + 6) = \frac{- 7776 + 45360 x^{\frac{1}{5}} + 38880 - 105840 x^{\frac{2}{5}} - 181440 x^{\frac{1}{5}} - 77760 + 123480 x^{\frac{3}{5}} + 317520 x^{\frac{2}{5}} + 272160 x^{\frac{1}{5}} + 77760 - 72030 x^{\frac{4}{5}} - 246960 x^{\frac{3}{5}} - 317520 x^{\frac{2}{5}} - 181440 x^{\frac{1}{5}} - 38880 + 16807 x + 5 (7^{4} {(x^{\frac{1}{5}})}^{4}) \cdot 6 + 10 {(7 x^{\frac{1}{5}})}^{3} \cdot 6^{2} + 10 {(7 x^{\frac{1}{5}})}^{2} \cdot 6^{3} + 5 (7 x^{\frac{1}{5}}) \cdot 6^{4} + 6^{5}}{16807}$

Schritt 1.5.2.4.18.6

Potenziere $7$ mit $4$ .

$f^{- 1} (7 x^{\frac{1}{5}} + 6) = \frac{- 7776 + 45360 x^{\frac{1}{5}} + 38880 - 105840 x^{\frac{2}{5}} - 181440 x^{\frac{1}{5}} - 77760 + 123480 x^{\frac{3}{5}} + 317520 x^{\frac{2}{5}} + 272160 x^{\frac{1}{5}} + 77760 - 72030 x^{\frac{4}{5}} - 246960 x^{\frac{3}{5}} - 317520 x^{\frac{2}{5}} - 181440 x^{\frac{1}{5}} - 38880 + 16807 x + 5 (2401 {(x^{\frac{1}{5}})}^{4}) \cdot 6 + 10 {(7 x^{\frac{1}{5}})}^{3} \cdot 6^{2} + 10 {(7 x^{\frac{1}{5}})}^{2} \cdot 6^{3} + 5 (7 x^{\frac{1}{5}}) \cdot 6^{4} + 6^{5}}{16807}$

Schritt 1.5.2.4.18.7

Multipliziere die Exponenten in ${(x^{\frac{1}{5}})}^{4}$ .

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Schritt 1.5.2.4.18.7.1

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

$f^{- 1} (7 x^{\frac{1}{5}} + 6) = \frac{- 7776 + 45360 x^{\frac{1}{5}} + 38880 - 105840 x^{\frac{2}{5}} - 181440 x^{\frac{1}{5}} - 77760 + 123480 x^{\frac{3}{5}} + 317520 x^{\frac{2}{5}} + 272160 x^{\frac{1}{5}} + 77760 - 72030 x^{\frac{4}{5}} - 246960 x^{\frac{3}{5}} - 317520 x^{\frac{2}{5}} - 181440 x^{\frac{1}{5}} - 38880 + 16807 x + 5 (2401 x^{\frac{1}{5} \cdot 4}) \cdot 6 + 10 {(7 x^{\frac{1}{5}})}^{3} \cdot 6^{2} + 10 {(7 x^{\frac{1}{5}})}^{2} \cdot 6^{3} + 5 (7 x^{\frac{1}{5}}) \cdot 6^{4} + 6^{5}}{16807}$

Schritt 1.5.2.4.18.7.2

Kombiniere $\frac{1}{5}$ und $4$ .

$f^{- 1} (7 x^{\frac{1}{5}} + 6) = \frac{- 7776 + 45360 x^{\frac{1}{5}} + 38880 - 105840 x^{\frac{2}{5}} - 181440 x^{\frac{1}{5}} - 77760 + 123480 x^{\frac{3}{5}} + 317520 x^{\frac{2}{5}} + 272160 x^{\frac{1}{5}} + 77760 - 72030 x^{\frac{4}{5}} - 246960 x^{\frac{3}{5}} - 317520 x^{\frac{2}{5}} - 181440 x^{\frac{1}{5}} - 38880 + 16807 x + 5 (2401 x^{\frac{4}{5}}) \cdot 6 + 10 {(7 x^{\frac{1}{5}})}^{3} \cdot 6^{2} + 10 {(7 x^{\frac{1}{5}})}^{2} \cdot 6^{3} + 5 (7 x^{\frac{1}{5}}) \cdot 6^{4} + 6^{5}}{16807}$

Schritt 1.5.2.4.18.8

Mutltipliziere $2401$ mit $5$ .

$f^{- 1} (7 x^{\frac{1}{5}} + 6) = \frac{- 7776 + 45360 x^{\frac{1}{5}} + 38880 - 105840 x^{\frac{2}{5}} - 181440 x^{\frac{1}{5}} - 77760 + 123480 x^{\frac{3}{5}} + 317520 x^{\frac{2}{5}} + 272160 x^{\frac{1}{5}} + 77760 - 72030 x^{\frac{4}{5}} - 246960 x^{\frac{3}{5}} - 317520 x^{\frac{2}{5}} - 181440 x^{\frac{1}{5}} - 38880 + 16807 x + 12005 x^{\frac{4}{5}} \cdot 6 + 10 {(7 x^{\frac{1}{5}})}^{3} \cdot 6^{2} + 10 {(7 x^{\frac{1}{5}})}^{2} \cdot 6^{3} + 5 (7 x^{\frac{1}{5}}) \cdot 6^{4} + 6^{5}}{16807}$

Schritt 1.5.2.4.18.9

Mutltipliziere $6$ mit $12005$ .

$f^{- 1} (7 x^{\frac{1}{5}} + 6) = \frac{- 7776 + 45360 x^{\frac{1}{5}} + 38880 - 105840 x^{\frac{2}{5}} - 181440 x^{\frac{1}{5}} - 77760 + 123480 x^{\frac{3}{5}} + 317520 x^{\frac{2}{5}} + 272160 x^{\frac{1}{5}} + 77760 - 72030 x^{\frac{4}{5}} - 246960 x^{\frac{3}{5}} - 317520 x^{\frac{2}{5}} - 181440 x^{\frac{1}{5}} - 38880 + 16807 x + 72030 x^{\frac{4}{5}} + 10 {(7 x^{\frac{1}{5}})}^{3} \cdot 6^{2} + 10 {(7 x^{\frac{1}{5}})}^{2} \cdot 6^{3} + 5 (7 x^{\frac{1}{5}}) \cdot 6^{4} + 6^{5}}{16807}$

Schritt 1.5.2.4.18.10

Wende die Produktregel auf $7 x^{\frac{1}{5}}$ an.

$f^{- 1} (7 x^{\frac{1}{5}} + 6) = \frac{- 7776 + 45360 x^{\frac{1}{5}} + 38880 - 105840 x^{\frac{2}{5}} - 181440 x^{\frac{1}{5}} - 77760 + 123480 x^{\frac{3}{5}} + 317520 x^{\frac{2}{5}} + 272160 x^{\frac{1}{5}} + 77760 - 72030 x^{\frac{4}{5}} - 246960 x^{\frac{3}{5}} - 317520 x^{\frac{2}{5}} - 181440 x^{\frac{1}{5}} - 38880 + 16807 x + 72030 x^{\frac{4}{5}} + 10 (7^{3} {(x^{\frac{1}{5}})}^{3}) \cdot 6^{2} + 10 {(7 x^{\frac{1}{5}})}^{2} \cdot 6^{3} + 5 (7 x^{\frac{1}{5}}) \cdot 6^{4} + 6^{5}}{16807}$

Schritt 1.5.2.4.18.11

Potenziere $7$ mit $3$ .

$f^{- 1} (7 x^{\frac{1}{5}} + 6) = \frac{- 7776 + 45360 x^{\frac{1}{5}} + 38880 - 105840 x^{\frac{2}{5}} - 181440 x^{\frac{1}{5}} - 77760 + 123480 x^{\frac{3}{5}} + 317520 x^{\frac{2}{5}} + 272160 x^{\frac{1}{5}} + 77760 - 72030 x^{\frac{4}{5}} - 246960 x^{\frac{3}{5}} - 317520 x^{\frac{2}{5}} - 181440 x^{\frac{1}{5}} - 38880 + 16807 x + 72030 x^{\frac{4}{5}} + 10 (343 {(x^{\frac{1}{5}})}^{3}) \cdot 6^{2} + 10 {(7 x^{\frac{1}{5}})}^{2} \cdot 6^{3} + 5 (7 x^{\frac{1}{5}}) \cdot 6^{4} + 6^{5}}{16807}$

Schritt 1.5.2.4.18.12

Multipliziere die Exponenten in ${(x^{\frac{1}{5}})}^{3}$ .

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Schritt 1.5.2.4.18.12.1

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

$f^{- 1} (7 x^{\frac{1}{5}} + 6) = \frac{- 7776 + 45360 x^{\frac{1}{5}} + 38880 - 105840 x^{\frac{2}{5}} - 181440 x^{\frac{1}{5}} - 77760 + 123480 x^{\frac{3}{5}} + 317520 x^{\frac{2}{5}} + 272160 x^{\frac{1}{5}} + 77760 - 72030 x^{\frac{4}{5}} - 246960 x^{\frac{3}{5}} - 317520 x^{\frac{2}{5}} - 181440 x^{\frac{1}{5}} - 38880 + 16807 x + 72030 x^{\frac{4}{5}} + 10 (343 x^{\frac{1}{5} \cdot 3}) \cdot 6^{2} + 10 {(7 x^{\frac{1}{5}})}^{2} \cdot 6^{3} + 5 (7 x^{\frac{1}{5}}) \cdot 6^{4} + 6^{5}}{16807}$

Schritt 1.5.2.4.18.12.2

Kombiniere $\frac{1}{5}$ und $3$ .

$f^{- 1} (7 x^{\frac{1}{5}} + 6) = \frac{- 7776 + 45360 x^{\frac{1}{5}} + 38880 - 105840 x^{\frac{2}{5}} - 181440 x^{\frac{1}{5}} - 77760 + 123480 x^{\frac{3}{5}} + 317520 x^{\frac{2}{5}} + 272160 x^{\frac{1}{5}} + 77760 - 72030 x^{\frac{4}{5}} - 246960 x^{\frac{3}{5}} - 317520 x^{\frac{2}{5}} - 181440 x^{\frac{1}{5}} - 38880 + 16807 x + 72030 x^{\frac{4}{5}} + 10 (343 x^{\frac{3}{5}}) \cdot 6^{2} + 10 {(7 x^{\frac{1}{5}})}^{2} \cdot 6^{3} + 5 (7 x^{\frac{1}{5}}) \cdot 6^{4} + 6^{5}}{16807}$

Schritt 1.5.2.4.18.13

Mutltipliziere $343$ mit $10$ .

$f^{- 1} (7 x^{\frac{1}{5}} + 6) = \frac{- 7776 + 45360 x^{\frac{1}{5}} + 38880 - 105840 x^{\frac{2}{5}} - 181440 x^{\frac{1}{5}} - 77760 + 123480 x^{\frac{3}{5}} + 317520 x^{\frac{2}{5}} + 272160 x^{\frac{1}{5}} + 77760 - 72030 x^{\frac{4}{5}} - 246960 x^{\frac{3}{5}} - 317520 x^{\frac{2}{5}} - 181440 x^{\frac{1}{5}} - 38880 + 16807 x + 72030 x^{\frac{4}{5}} + 3430 x^{\frac{3}{5}} \cdot 6^{2} + 10 {(7 x^{\frac{1}{5}})}^{2} \cdot 6^{3} + 5 (7 x^{\frac{1}{5}}) \cdot 6^{4} + 6^{5}}{16807}$

Schritt 1.5.2.4.18.14

Potenziere $6$ mit $2$ .

$f^{- 1} (7 x^{\frac{1}{5}} + 6) = \frac{- 7776 + 45360 x^{\frac{1}{5}} + 38880 - 105840 x^{\frac{2}{5}} - 181440 x^{\frac{1}{5}} - 77760 + 123480 x^{\frac{3}{5}} + 317520 x^{\frac{2}{5}} + 272160 x^{\frac{1}{5}} + 77760 - 72030 x^{\frac{4}{5}} - 246960 x^{\frac{3}{5}} - 317520 x^{\frac{2}{5}} - 181440 x^{\frac{1}{5}} - 38880 + 16807 x + 72030 x^{\frac{4}{5}} + 3430 x^{\frac{3}{5}} \cdot 36 + 10 {(7 x^{\frac{1}{5}})}^{2} \cdot 6^{3} + 5 (7 x^{\frac{1}{5}}) \cdot 6^{4} + 6^{5}}{16807}$

Schritt 1.5.2.4.18.15

Mutltipliziere $36$ mit $3430$ .

$f^{- 1} (7 x^{\frac{1}{5}} + 6) = \frac{- 7776 + 45360 x^{\frac{1}{5}} + 38880 - 105840 x^{\frac{2}{5}} - 181440 x^{\frac{1}{5}} - 77760 + 123480 x^{\frac{3}{5}} + 317520 x^{\frac{2}{5}} + 272160 x^{\frac{1}{5}} + 77760 - 72030 x^{\frac{4}{5}} - 246960 x^{\frac{3}{5}} - 317520 x^{\frac{2}{5}} - 181440 x^{\frac{1}{5}} - 38880 + 16807 x + 72030 x^{\frac{4}{5}} + 123480 x^{\frac{3}{5}} + 10 {(7 x^{\frac{1}{5}})}^{2} \cdot 6^{3} + 5 (7 x^{\frac{1}{5}}) \cdot 6^{4} + 6^{5}}{16807}$

Schritt 1.5.2.4.18.16

Wende die Produktregel auf $7 x^{\frac{1}{5}}$ an.

$f^{- 1} (7 x^{\frac{1}{5}} + 6) = \frac{- 7776 + 45360 x^{\frac{1}{5}} + 38880 - 105840 x^{\frac{2}{5}} - 181440 x^{\frac{1}{5}} - 77760 + 123480 x^{\frac{3}{5}} + 317520 x^{\frac{2}{5}} + 272160 x^{\frac{1}{5}} + 77760 - 72030 x^{\frac{4}{5}} - 246960 x^{\frac{3}{5}} - 317520 x^{\frac{2}{5}} - 181440 x^{\frac{1}{5}} - 38880 + 16807 x + 72030 x^{\frac{4}{5}} + 123480 x^{\frac{3}{5}} + 10 (7^{2} {(x^{\frac{1}{5}})}^{2}) \cdot 6^{3} + 5 (7 x^{\frac{1}{5}}) \cdot 6^{4} + 6^{5}}{16807}$

Schritt 1.5.2.4.18.17

Potenziere $7$ mit $2$ .

$f^{- 1} (7 x^{\frac{1}{5}} + 6) = \frac{- 7776 + 45360 x^{\frac{1}{5}} + 38880 - 105840 x^{\frac{2}{5}} - 181440 x^{\frac{1}{5}} - 77760 + 123480 x^{\frac{3}{5}} + 317520 x^{\frac{2}{5}} + 272160 x^{\frac{1}{5}} + 77760 - 72030 x^{\frac{4}{5}} - 246960 x^{\frac{3}{5}} - 317520 x^{\frac{2}{5}} - 181440 x^{\frac{1}{5}} - 38880 + 16807 x + 72030 x^{\frac{4}{5}} + 123480 x^{\frac{3}{5}} + 10 (49 {(x^{\frac{1}{5}})}^{2}) \cdot 6^{3} + 5 (7 x^{\frac{1}{5}}) \cdot 6^{4} + 6^{5}}{16807}$

Schritt 1.5.2.4.18.18

Multipliziere die Exponenten in ${(x^{\frac{1}{5}})}^{2}$ .

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Schritt 1.5.2.4.18.18.1

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

$f^{- 1} (7 x^{\frac{1}{5}} + 6) = \frac{- 7776 + 45360 x^{\frac{1}{5}} + 38880 - 105840 x^{\frac{2}{5}} - 181440 x^{\frac{1}{5}} - 77760 + 123480 x^{\frac{3}{5}} + 317520 x^{\frac{2}{5}} + 272160 x^{\frac{1}{5}} + 77760 - 72030 x^{\frac{4}{5}} - 246960 x^{\frac{3}{5}} - 317520 x^{\frac{2}{5}} - 181440 x^{\frac{1}{5}} - 38880 + 16807 x + 72030 x^{\frac{4}{5}} + 123480 x^{\frac{3}{5}} + 10 (49 x^{\frac{1}{5} \cdot 2}) \cdot 6^{3} + 5 (7 x^{\frac{1}{5}}) \cdot 6^{4} + 6^{5}}{16807}$

Schritt 1.5.2.4.18.18.2

Kombiniere $\frac{1}{5}$ und $2$ .

$f^{- 1} (7 x^{\frac{1}{5}} + 6) = \frac{- 7776 + 45360 x^{\frac{1}{5}} + 38880 - 105840 x^{\frac{2}{5}} - 181440 x^{\frac{1}{5}} - 77760 + 123480 x^{\frac{3}{5}} + 317520 x^{\frac{2}{5}} + 272160 x^{\frac{1}{5}} + 77760 - 72030 x^{\frac{4}{5}} - 246960 x^{\frac{3}{5}} - 317520 x^{\frac{2}{5}} - 181440 x^{\frac{1}{5}} - 38880 + 16807 x + 72030 x^{\frac{4}{5}} + 123480 x^{\frac{3}{5}} + 10 (49 x^{\frac{2}{5}}) \cdot 6^{3} + 5 (7 x^{\frac{1}{5}}) \cdot 6^{4} + 6^{5}}{16807}$

Schritt 1.5.2.4.18.19

Mutltipliziere $49$ mit $10$ .

$f^{- 1} (7 x^{\frac{1}{5}} + 6) = \frac{- 7776 + 45360 x^{\frac{1}{5}} + 38880 - 105840 x^{\frac{2}{5}} - 181440 x^{\frac{1}{5}} - 77760 + 123480 x^{\frac{3}{5}} + 317520 x^{\frac{2}{5}} + 272160 x^{\frac{1}{5}} + 77760 - 72030 x^{\frac{4}{5}} - 246960 x^{\frac{3}{5}} - 317520 x^{\frac{2}{5}} - 181440 x^{\frac{1}{5}} - 38880 + 16807 x + 72030 x^{\frac{4}{5}} + 123480 x^{\frac{3}{5}} + 490 x^{\frac{2}{5}} \cdot 6^{3} + 5 (7 x^{\frac{1}{5}}) \cdot 6^{4} + 6^{5}}{16807}$

Schritt 1.5.2.4.18.20

Potenziere $6$ mit $3$ .

$f^{- 1} (7 x^{\frac{1}{5}} + 6) = \frac{- 7776 + 45360 x^{\frac{1}{5}} + 38880 - 105840 x^{\frac{2}{5}} - 181440 x^{\frac{1}{5}} - 77760 + 123480 x^{\frac{3}{5}} + 317520 x^{\frac{2}{5}} + 272160 x^{\frac{1}{5}} + 77760 - 72030 x^{\frac{4}{5}} - 246960 x^{\frac{3}{5}} - 317520 x^{\frac{2}{5}} - 181440 x^{\frac{1}{5}} - 38880 + 16807 x + 72030 x^{\frac{4}{5}} + 123480 x^{\frac{3}{5}} + 490 x^{\frac{2}{5}} \cdot 216 + 5 (7 x^{\frac{1}{5}}) \cdot 6^{4} + 6^{5}}{16807}$

Schritt 1.5.2.4.18.21

Mutltipliziere $216$ mit $490$ .

$f^{- 1} (7 x^{\frac{1}{5}} + 6) = \frac{- 7776 + 45360 x^{\frac{1}{5}} + 38880 - 105840 x^{\frac{2}{5}} - 181440 x^{\frac{1}{5}} - 77760 + 123480 x^{\frac{3}{5}} + 317520 x^{\frac{2}{5}} + 272160 x^{\frac{1}{5}} + 77760 - 72030 x^{\frac{4}{5}} - 246960 x^{\frac{3}{5}} - 317520 x^{\frac{2}{5}} - 181440 x^{\frac{1}{5}} - 38880 + 16807 x + 72030 x^{\frac{4}{5}} + 123480 x^{\frac{3}{5}} + 105840 x^{\frac{2}{5}} + 5 (7 x^{\frac{1}{5}}) \cdot 6^{4} + 6^{5}}{16807}$

Schritt 1.5.2.4.18.22

Mutltipliziere $7$ mit $5$ .

$f^{- 1} (7 x^{\frac{1}{5}} + 6) = \frac{- 7776 + 45360 x^{\frac{1}{5}} + 38880 - 105840 x^{\frac{2}{5}} - 181440 x^{\frac{1}{5}} - 77760 + 123480 x^{\frac{3}{5}} + 317520 x^{\frac{2}{5}} + 272160 x^{\frac{1}{5}} + 77760 - 72030 x^{\frac{4}{5}} - 246960 x^{\frac{3}{5}} - 317520 x^{\frac{2}{5}} - 181440 x^{\frac{1}{5}} - 38880 + 16807 x + 72030 x^{\frac{4}{5}} + 123480 x^{\frac{3}{5}} + 105840 x^{\frac{2}{5}} + 35 x^{\frac{1}{5}} \cdot 6^{4} + 6^{5}}{16807}$

Schritt 1.5.2.4.18.23

Potenziere $6$ mit $4$ .

$f^{- 1} (7 x^{\frac{1}{5}} + 6) = \frac{- 7776 + 45360 x^{\frac{1}{5}} + 38880 - 105840 x^{\frac{2}{5}} - 181440 x^{\frac{1}{5}} - 77760 + 123480 x^{\frac{3}{5}} + 317520 x^{\frac{2}{5}} + 272160 x^{\frac{1}{5}} + 77760 - 72030 x^{\frac{4}{5}} - 246960 x^{\frac{3}{5}} - 317520 x^{\frac{2}{5}} - 181440 x^{\frac{1}{5}} - 38880 + 16807 x + 72030 x^{\frac{4}{5}} + 123480 x^{\frac{3}{5}} + 105840 x^{\frac{2}{5}} + 35 x^{\frac{1}{5}} \cdot 1296 + 6^{5}}{16807}$

Schritt 1.5.2.4.18.24

Mutltipliziere $1296$ mit $35$ .

$f^{- 1} (7 x^{\frac{1}{5}} + 6) = \frac{- 7776 + 45360 x^{\frac{1}{5}} + 38880 - 105840 x^{\frac{2}{5}} - 181440 x^{\frac{1}{5}} - 77760 + 123480 x^{\frac{3}{5}} + 317520 x^{\frac{2}{5}} + 272160 x^{\frac{1}{5}} + 77760 - 72030 x^{\frac{4}{5}} - 246960 x^{\frac{3}{5}} - 317520 x^{\frac{2}{5}} - 181440 x^{\frac{1}{5}} - 38880 + 16807 x + 72030 x^{\frac{4}{5}} + 123480 x^{\frac{3}{5}} + 105840 x^{\frac{2}{5}} + 45360 x^{\frac{1}{5}} + 6^{5}}{16807}$

Schritt 1.5.2.4.18.25

Potenziere $6$ mit $5$ .

$f^{- 1} (7 x^{\frac{1}{5}} + 6) = \frac{- 7776 + 45360 x^{\frac{1}{5}} + 38880 - 105840 x^{\frac{2}{5}} - 181440 x^{\frac{1}{5}} - 77760 + 123480 x^{\frac{3}{5}} + 317520 x^{\frac{2}{5}} + 272160 x^{\frac{1}{5}} + 77760 - 72030 x^{\frac{4}{5}} - 246960 x^{\frac{3}{5}} - 317520 x^{\frac{2}{5}} - 181440 x^{\frac{1}{5}} - 38880 + 16807 x + 72030 x^{\frac{4}{5}} + 123480 x^{\frac{3}{5}} + 105840 x^{\frac{2}{5}} + 45360 x^{\frac{1}{5}} + 7776}{16807}$

Schritt 1.5.2.5

Vereinfache Terme.

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Schritt 1.5.2.5.1

Vereine die Terme mit entgegengesetztem Vorzeichen in $- 7776 + 45360 x^{\frac{1}{5}} + 38880 - 105840 x^{\frac{2}{5}} - 181440 x^{\frac{1}{5}} - 77760 + 123480 x^{\frac{3}{5}} + 317520 x^{\frac{2}{5}} + 272160 x^{\frac{1}{5}} + 77760 - 72030 x^{\frac{4}{5}} - 246960 x^{\frac{3}{5}} - 317520 x^{\frac{2}{5}} - 181440 x^{\frac{1}{5}} - 38880 + 16807 x + 72030 x^{\frac{4}{5}} + 123480 x^{\frac{3}{5}} + 105840 x^{\frac{2}{5}} + 45360 x^{\frac{1}{5}} + 7776$ .

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Schritt 1.5.2.5.1.1

Addiere $- 77760$ und $77760$ .

$f^{- 1} (7 x^{\frac{1}{5}} + 6) = \frac{- 7776 + 45360 x^{\frac{1}{5}} + 38880 - 105840 x^{\frac{2}{5}} - 181440 x^{\frac{1}{5}} + 123480 x^{\frac{3}{5}} + 317520 x^{\frac{2}{5}} + 272160 x^{\frac{1}{5}} + 0 - 72030 x^{\frac{4}{5}} - 246960 x^{\frac{3}{5}} - 317520 x^{\frac{2}{5}} - 181440 x^{\frac{1}{5}} - 38880 + 16807 x + 72030 x^{\frac{4}{5}} + 123480 x^{\frac{3}{5}} + 105840 x^{\frac{2}{5}} + 45360 x^{\frac{1}{5}} + 7776}{16807}$

Schritt 1.5.2.5.1.2

Addiere $- 7776 + 45360 x^{\frac{1}{5}} + 38880 - 105840 x^{\frac{2}{5}} - 181440 x^{\frac{1}{5}} + 123480 x^{\frac{3}{5}} + 317520 x^{\frac{2}{5}} + 272160 x^{\frac{1}{5}}$ und $0$ .

$f^{- 1} (7 x^{\frac{1}{5}} + 6) = \frac{- 7776 + 45360 x^{\frac{1}{5}} + 38880 - 105840 x^{\frac{2}{5}} - 181440 x^{\frac{1}{5}} + 123480 x^{\frac{3}{5}} + 317520 x^{\frac{2}{5}} + 272160 x^{\frac{1}{5}} - 72030 x^{\frac{4}{5}} - 246960 x^{\frac{3}{5}} - 317520 x^{\frac{2}{5}} - 181440 x^{\frac{1}{5}} - 38880 + 16807 x + 72030 x^{\frac{4}{5}} + 123480 x^{\frac{3}{5}} + 105840 x^{\frac{2}{5}} + 45360 x^{\frac{1}{5}} + 7776}{16807}$

Schritt 1.5.2.5.1.3

Subtrahiere $317520 x^{\frac{2}{5}}$ von $317520 x^{\frac{2}{5}}$ .

$f^{- 1} (7 x^{\frac{1}{5}} + 6) = \frac{- 7776 + 45360 x^{\frac{1}{5}} + 38880 - 105840 x^{\frac{2}{5}} - 181440 x^{\frac{1}{5}} + 123480 x^{\frac{3}{5}} + 272160 x^{\frac{1}{5}} - 72030 x^{\frac{4}{5}} - 246960 x^{\frac{3}{5}} + 0 - 181440 x^{\frac{1}{5}} - 38880 + 16807 x + 72030 x^{\frac{4}{5}} + 123480 x^{\frac{3}{5}} + 105840 x^{\frac{2}{5}} + 45360 x^{\frac{1}{5}} + 7776}{16807}$

Schritt 1.5.2.5.1.4

Addiere $- 7776 + 45360 x^{\frac{1}{5}} + 38880 - 105840 x^{\frac{2}{5}} - 181440 x^{\frac{1}{5}} + 123480 x^{\frac{3}{5}} + 272160 x^{\frac{1}{5}} - 72030 x^{\frac{4}{5}} - 246960 x^{\frac{3}{5}}$ und $0$ .

$f^{- 1} (7 x^{\frac{1}{5}} + 6) = \frac{- 7776 + 45360 x^{\frac{1}{5}} + 38880 - 105840 x^{\frac{2}{5}} - 181440 x^{\frac{1}{5}} + 123480 x^{\frac{3}{5}} + 272160 x^{\frac{1}{5}} - 72030 x^{\frac{4}{5}} - 246960 x^{\frac{3}{5}} - 181440 x^{\frac{1}{5}} - 38880 + 16807 x + 72030 x^{\frac{4}{5}} + 123480 x^{\frac{3}{5}} + 105840 x^{\frac{2}{5}} + 45360 x^{\frac{1}{5}} + 7776}{16807}$

Schritt 1.5.2.5.1.5

Subtrahiere $38880$ von $38880$ .

$f^{- 1} (7 x^{\frac{1}{5}} + 6) = \frac{- 7776 + 45360 x^{\frac{1}{5}} - 105840 x^{\frac{2}{5}} - 181440 x^{\frac{1}{5}} + 123480 x^{\frac{3}{5}} + 272160 x^{\frac{1}{5}} - 72030 x^{\frac{4}{5}} - 246960 x^{\frac{3}{5}} - 181440 x^{\frac{1}{5}} + 0 + 16807 x + 72030 x^{\frac{4}{5}} + 123480 x^{\frac{3}{5}} + 105840 x^{\frac{2}{5}} + 45360 x^{\frac{1}{5}} + 7776}{16807}$

Schritt 1.5.2.5.1.6

Addiere $- 7776 + 45360 x^{\frac{1}{5}} - 105840 x^{\frac{2}{5}} - 181440 x^{\frac{1}{5}} + 123480 x^{\frac{3}{5}} + 272160 x^{\frac{1}{5}} - 72030 x^{\frac{4}{5}} - 246960 x^{\frac{3}{5}} - 181440 x^{\frac{1}{5}}$ und $0$ .

$f^{- 1} (7 x^{\frac{1}{5}} + 6) = \frac{- 7776 + 45360 x^{\frac{1}{5}} - 105840 x^{\frac{2}{5}} - 181440 x^{\frac{1}{5}} + 123480 x^{\frac{3}{5}} + 272160 x^{\frac{1}{5}} - 72030 x^{\frac{4}{5}} - 246960 x^{\frac{3}{5}} - 181440 x^{\frac{1}{5}} + 16807 x + 72030 x^{\frac{4}{5}} + 123480 x^{\frac{3}{5}} + 105840 x^{\frac{2}{5}} + 45360 x^{\frac{1}{5}} + 7776}{16807}$

Schritt 1.5.2.5.1.7

Addiere $- 72030 x^{\frac{4}{5}}$ und $72030 x^{\frac{4}{5}}$ .

$f^{- 1} (7 x^{\frac{1}{5}} + 6) = \frac{- 7776 + 45360 x^{\frac{1}{5}} - 105840 x^{\frac{2}{5}} - 181440 x^{\frac{1}{5}} + 123480 x^{\frac{3}{5}} + 272160 x^{\frac{1}{5}} - 246960 x^{\frac{3}{5}} - 181440 x^{\frac{1}{5}} + 16807 x + 0 + 123480 x^{\frac{3}{5}} + 105840 x^{\frac{2}{5}} + 45360 x^{\frac{1}{5}} + 7776}{16807}$

Schritt 1.5.2.5.1.8

Addiere $- 7776 + 45360 x^{\frac{1}{5}} - 105840 x^{\frac{2}{5}} - 181440 x^{\frac{1}{5}} + 123480 x^{\frac{3}{5}} + 272160 x^{\frac{1}{5}} - 246960 x^{\frac{3}{5}} - 181440 x^{\frac{1}{5}} + 16807 x$ und $0$ .

$f^{- 1} (7 x^{\frac{1}{5}} + 6) = \frac{- 7776 + 45360 x^{\frac{1}{5}} - 105840 x^{\frac{2}{5}} - 181440 x^{\frac{1}{5}} + 123480 x^{\frac{3}{5}} + 272160 x^{\frac{1}{5}} - 246960 x^{\frac{3}{5}} - 181440 x^{\frac{1}{5}} + 16807 x + 123480 x^{\frac{3}{5}} + 105840 x^{\frac{2}{5}} + 45360 x^{\frac{1}{5}} + 7776}{16807}$

Schritt 1.5.2.5.1.9

Addiere $- 105840 x^{\frac{2}{5}}$ und $105840 x^{\frac{2}{5}}$ .

$f^{- 1} (7 x^{\frac{1}{5}} + 6) = \frac{- 7776 + 45360 x^{\frac{1}{5}} - 181440 x^{\frac{1}{5}} + 123480 x^{\frac{3}{5}} + 272160 x^{\frac{1}{5}} - 246960 x^{\frac{3}{5}} - 181440 x^{\frac{1}{5}} + 16807 x + 123480 x^{\frac{3}{5}} + 0 + 45360 x^{\frac{1}{5}} + 7776}{16807}$

Schritt 1.5.2.5.1.10

Addiere $- 7776 + 45360 x^{\frac{1}{5}} - 181440 x^{\frac{1}{5}} + 123480 x^{\frac{3}{5}} + 272160 x^{\frac{1}{5}} - 246960 x^{\frac{3}{5}} - 181440 x^{\frac{1}{5}} + 16807 x + 123480 x^{\frac{3}{5}}$ und $0$ .

Schritt 1.5.2.5.1.11

Addiere $- 7776$ und $7776$ .

$f^{- 1} (7 x^{\frac{1}{5}} + 6) = \frac{45360 x^{\frac{1}{5}} - 181440 x^{\frac{1}{5}} + 123480 x^{\frac{3}{5}} + 272160 x^{\frac{1}{5}} - 246960 x^{\frac{3}{5}} - 181440 x^{\frac{1}{5}} + 16807 x + 123480 x^{\frac{3}{5}} + 45360 x^{\frac{1}{5}} + 0}{16807}$

Schritt 1.5.2.5.1.12

Addiere $45360 x^{\frac{1}{5}} - 181440 x^{\frac{1}{5}} + 123480 x^{\frac{3}{5}} + 272160 x^{\frac{1}{5}} - 246960 x^{\frac{3}{5}} - 181440 x^{\frac{1}{5}} + 16807 x + 123480 x^{\frac{3}{5}} + 45360 x^{\frac{1}{5}}$ und $0$ .

Schritt 1.5.2.5.2

Subtrahiere $181440 x^{\frac{1}{5}}$ von $45360 x^{\frac{1}{5}}$ .

$f^{- 1} (7 x^{\frac{1}{5}} + 6) = \frac{- 136080 x^{\frac{1}{5}} + 123480 x^{\frac{3}{5}} + 272160 x^{\frac{1}{5}} - 246960 x^{\frac{3}{5}} - 181440 x^{\frac{1}{5}} + 16807 x + 123480 x^{\frac{3}{5}} + 45360 x^{\frac{1}{5}}}{16807}$

Schritt 1.5.2.5.3

Addiere $- 136080 x^{\frac{1}{5}}$ und $272160 x^{\frac{1}{5}}$ .

$f^{- 1} (7 x^{\frac{1}{5}} + 6) = \frac{136080 x^{\frac{1}{5}} + 123480 x^{\frac{3}{5}} - 246960 x^{\frac{3}{5}} - 181440 x^{\frac{1}{5}} + 16807 x + 123480 x^{\frac{3}{5}} + 45360 x^{\frac{1}{5}}}{16807}$

Schritt 1.5.2.5.4

Subtrahiere $181440 x^{\frac{1}{5}}$ von $136080 x^{\frac{1}{5}}$ .

$f^{- 1} (7 x^{\frac{1}{5}} + 6) = \frac{- 45360 x^{\frac{1}{5}} + 123480 x^{\frac{3}{5}} - 246960 x^{\frac{3}{5}} + 16807 x + 123480 x^{\frac{3}{5}} + 45360 x^{\frac{1}{5}}}{16807}$

Schritt 1.5.2.5.5

Vereine die Terme mit entgegengesetztem Vorzeichen in $- 45360 x^{\frac{1}{5}} + 123480 x^{\frac{3}{5}} - 246960 x^{\frac{3}{5}} + 16807 x + 123480 x^{\frac{3}{5}} + 45360 x^{\frac{1}{5}}$ .

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Schritt 1.5.2.5.5.1

Addiere $- 45360 x^{\frac{1}{5}}$ und $45360 x^{\frac{1}{5}}$ .

$f^{- 1} (7 x^{\frac{1}{5}} + 6) = \frac{123480 x^{\frac{3}{5}} - 246960 x^{\frac{3}{5}} + 16807 x + 123480 x^{\frac{3}{5}} + 0}{16807}$

Schritt 1.5.2.5.5.2

Addiere $123480 x^{\frac{3}{5}} - 246960 x^{\frac{3}{5}} + 16807 x + 123480 x^{\frac{3}{5}}$ und $0$ .

$f^{- 1} (7 x^{\frac{1}{5}} + 6) = \frac{123480 x^{\frac{3}{5}} - 246960 x^{\frac{3}{5}} + 16807 x + 123480 x^{\frac{3}{5}}}{16807}$

Schritt 1.5.2.5.6

Subtrahiere $246960 x^{\frac{3}{5}}$ von $123480 x^{\frac{3}{5}}$ .

$f^{- 1} (7 x^{\frac{1}{5}} + 6) = \frac{- 123480 x^{\frac{3}{5}} + 16807 x + 123480 x^{\frac{3}{5}}}{16807}$

Schritt 1.5.2.5.7

Vereine die Terme mit entgegengesetztem Vorzeichen in $- 123480 x^{\frac{3}{5}} + 16807 x + 123480 x^{\frac{3}{5}}$ .

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Schritt 1.5.2.5.7.1

Addiere $- 123480 x^{\frac{3}{5}}$ und $123480 x^{\frac{3}{5}}$ .

$f^{- 1} (7 x^{\frac{1}{5}} + 6) = \frac{16807 x + 0}{16807}$

Schritt 1.5.2.5.7.2

Addiere $16807 x$ und $0$ .

$f^{- 1} (7 x^{\frac{1}{5}} + 6) = \frac{16807 x}{16807}$

Schritt 1.5.2.5.8

Kürze den gemeinsamen Faktor von $16807$ .

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Schritt 1.5.2.5.8.1

Kürze den gemeinsamen Faktor.

$f^{- 1} (7 x^{\frac{1}{5}} + 6) = \frac{16807 x}{16807}$

Schritt 1.5.2.5.8.2

Dividiere $x$ durch $1$ .

$f^{- 1} (7 x^{\frac{1}{5}} + 6) = x$

Schritt 1.5.3

Berechne $f (f^{- 1} (x))$ .

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Schritt 1.5.3.1

Bilde die verkettete Ergebnisfunktion.

$f (f^{- 1} (x))$

Schritt 1.5.3.2

Berechne $f (- \frac{7776}{16807} + \frac{6480 x}{16807} - \frac{2160 x^{2}}{16807} + \frac{360 x^{3}}{16807} - \frac{30 x^{4}}{16807} + \frac{x^{5}}{16807})$ durch Einsetzen des Wertes von $f^{- 1}$ in $f$ .

$f (- \frac{7776}{16807} + \frac{6480 x}{16807} - \frac{2160 x^{2}}{16807} + \frac{360 x^{3}}{16807} - \frac{30 x^{4}}{16807} + \frac{x^{5}}{16807}) = 7 {(- \frac{7776}{16807} + \frac{6480 x}{16807} - \frac{2160 x^{2}}{16807} + \frac{360 x^{3}}{16807} - \frac{30 x^{4}}{16807} + \frac{x^{5}}{16807})}^{\frac{1}{5}} + 6$

Schritt 1.5.3.3

Bewege $- \frac{7776}{16807}$ .

$f (- \frac{7776}{16807} + \frac{6480 x}{16807} - \frac{2160 x^{2}}{16807} + \frac{360 x^{3}}{16807} - \frac{30 x^{4}}{16807} + \frac{x^{5}}{16807}) = 7 {(\frac{6480 x}{16807} - \frac{2160 x^{2}}{16807} + \frac{360 x^{3}}{16807} - \frac{30 x^{4}}{16807} + \frac{x^{5}}{16807} - \frac{7776}{16807})}^{\frac{1}{5}} + 6$

Schritt 1.5.3.4

Bewege $\frac{6480 x}{16807}$ .

$f (- \frac{7776}{16807} + \frac{6480 x}{16807} - \frac{2160 x^{2}}{16807} + \frac{360 x^{3}}{16807} - \frac{30 x^{4}}{16807} + \frac{x^{5}}{16807}) = 7 {(- \frac{2160 x^{2}}{16807} + \frac{360 x^{3}}{16807} - \frac{30 x^{4}}{16807} + \frac{x^{5}}{16807} + \frac{6480 x}{16807} - \frac{7776}{16807})}^{\frac{1}{5}} + 6$

Schritt 1.5.3.5

Bewege $- \frac{2160 x^{2}}{16807}$ .

$f (- \frac{7776}{16807} + \frac{6480 x}{16807} - \frac{2160 x^{2}}{16807} + \frac{360 x^{3}}{16807} - \frac{30 x^{4}}{16807} + \frac{x^{5}}{16807}) = 7 {(\frac{360 x^{3}}{16807} - \frac{30 x^{4}}{16807} + \frac{x^{5}}{16807} - \frac{2160 x^{2}}{16807} + \frac{6480 x}{16807} - \frac{7776}{16807})}^{\frac{1}{5}} + 6$

Schritt 1.5.3.6

Bewege $\frac{360 x^{3}}{16807}$ .

$f (- \frac{7776}{16807} + \frac{6480 x}{16807} - \frac{2160 x^{2}}{16807} + \frac{360 x^{3}}{16807} - \frac{30 x^{4}}{16807} + \frac{x^{5}}{16807}) = 7 {(- \frac{30 x^{4}}{16807} + \frac{x^{5}}{16807} + \frac{360 x^{3}}{16807} - \frac{2160 x^{2}}{16807} + \frac{6480 x}{16807} - \frac{7776}{16807})}^{\frac{1}{5}} + 6$

Schritt 1.5.3.7

Stelle $- \frac{30 x^{4}}{16807}$ und $\frac{x^{5}}{16807}$ um.

$f (- \frac{7776}{16807} + \frac{6480 x}{16807} - \frac{2160 x^{2}}{16807} + \frac{360 x^{3}}{16807} - \frac{30 x^{4}}{16807} + \frac{x^{5}}{16807}) = 7 {(\frac{x^{5}}{16807} - \frac{30 x^{4}}{16807} + \frac{360 x^{3}}{16807} - \frac{2160 x^{2}}{16807} + \frac{6480 x}{16807} - \frac{7776}{16807})}^{\frac{1}{5}} + 6$

Schritt 1.5.4

Da $f^{- 1} (f (x)) = x$ und $f (f^{- 1} (x)) = x$ gleich sind, ist $f^{- 1} (x) = - \frac{7776}{16807} + \frac{6480 x}{16807} - \frac{2160 x^{2}}{16807} + \frac{360 x^{3}}{16807} - \frac{30 x^{4}}{16807} + \frac{x^{5}}{16807}$ die inverse Funktion (Umkehrfunktion) von $f (x) = 7 x^{\frac{1}{5}} + 6$ .

$f^{- 1} (x) = - \frac{7776}{16807} + \frac{6480 x}{16807} - \frac{2160 x^{2}}{16807} + \frac{360 x^{3}}{16807} - \frac{30 x^{4}}{16807} + \frac{x^{5}}{16807}$

Schritt 2

Vereinfache $- \frac{7776}{16807} + \frac{6480 x}{16807} - \frac{2160 x^{2}}{16807} + \frac{360 x^{3}}{16807} - \frac{30 x^{4}}{16807} + \frac{x^{5}}{16807}$ .

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Schritt 2.1

Bewege $- \frac{7776}{16807}$ .

$\frac{6480 x}{16807} - \frac{2160 x^{2}}{16807} + \frac{360 x^{3}}{16807} - \frac{30 x^{4}}{16807} + \frac{x^{5}}{16807} - \frac{7776}{16807}$

Schritt 2.2

Bewege $\frac{6480 x}{16807}$ .

$- \frac{2160 x^{2}}{16807} + \frac{360 x^{3}}{16807} - \frac{30 x^{4}}{16807} + \frac{x^{5}}{16807} + \frac{6480 x}{16807} - \frac{7776}{16807}$

Schritt 2.3

Bewege $- \frac{2160 x^{2}}{16807}$ .

$\frac{360 x^{3}}{16807} - \frac{30 x^{4}}{16807} + \frac{x^{5}}{16807} - \frac{2160 x^{2}}{16807} + \frac{6480 x}{16807} - \frac{7776}{16807}$

Schritt 2.4

Bewege $\frac{360 x^{3}}{16807}$ .

$- \frac{30 x^{4}}{16807} + \frac{x^{5}}{16807} + \frac{360 x^{3}}{16807} - \frac{2160 x^{2}}{16807} + \frac{6480 x}{16807} - \frac{7776}{16807}$

Schritt 2.5

Stelle $- \frac{30 x^{4}}{16807}$ und $\frac{x^{5}}{16807}$ um.

$\frac{x^{5}}{16807} - \frac{30 x^{4}}{16807} + \frac{360 x^{3}}{16807} - \frac{2160 x^{2}}{16807} + \frac{6480 x}{16807} - \frac{7776}{16807}$