vereinfachen \sqrt[4]{((7a^5b^{-6}-c)/(16b^3c))^5}

Lösung

vereinfachen

4 (\frac{7 a ^{5} b ^{- 6} - c}{16 b ^{3} c})^{5}

\frac{( 7 a ^{5} - b ^{6} c ) ^{\frac{5}{4}}}{32 b ^{\frac{45}{4}} c ^{\frac{5}{4}}}

Schritte zur Lösung

4 (\frac{7 a ^{5} b ^{- 6} - c}{16 b ^{3} c})^{5}

Wende Radikal Regel an:

n a = a^{\frac{1}{n}}

= ((\frac{7 a ^{5} b ^{- 6} - c}{16 b ^{3} c})^{5})^{\frac{1}{4}}

Wende Exponentenregel an:

(a^{b})^{c} = a^{b c},

angenommen

a \geq 0

= (\frac{7 a ^{5} b ^{- 6} - c}{16 b ^{3} c})^{5 \cdot \frac{1}{4}}

5 \cdot \frac{1}{4} = \frac{5}{4}

= (\frac{7 a ^{5} b ^{- 6} - c}{16 b ^{3} c})^{\frac{5}{4}}

\frac{7 a ^{5} b ^{- 6} - c}{16 b ^{3} c} = \frac{7 a ^{5} - b ^{6} c}{16 b ^{9} c}

= (\frac{7 a ^{5} - c b ^{6}}{16 b ^{9} c})^{\frac{5}{4}}

Wende Exponentenregel an:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

= \frac{( 7 a ^{5} - b ^{6} c ) ^{\frac{5}{4}}}{( 16 b ^{9} c ) ^{\frac{5}{4}}}

Wende Exponentenregel an:

(a \cdot b)^{n} = a^{n} b^{n}

(16 b^{9} c)^{\frac{5}{4}} = 1 6^{\frac{5}{4}} c^{\frac{5}{4}} (b^{9})^{\frac{5}{4}}

= \frac{( 7 a ^{5} - b ^{6} c ) ^{\frac{5}{4}}}{1 6 ^{\frac{5}{4}} c ^{\frac{5}{4}} ( b ^{9} ) ^{\frac{5}{4}}}

1 6^{\frac{5}{4}} = 32

= \frac{( 7 a ^{5} - b ^{6} c ) ^{\frac{5}{4}}}{32 c ^{\frac{5}{4}} ( b ^{9} ) ^{\frac{5}{4}}}

(b^{9})^{\frac{5}{4}} : b^{\frac{45}{4}}

= \frac{( 7 a ^{5} - c b ^{6} ) ^{\frac{5}{4}}}{32 b ^{\frac{45}{4}} c ^{\frac{5}{4}}}

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