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

The solution to \sqrt[4]{((7a^5b^{-6}-c)/(16b^3c))^5} is ((7a^5-b^6c)^{5/4})/(32b^{45/4)c^{5/4}}

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

simplify

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

Solution steps

Apply radical rule:

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

Apply exponent rule:

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

assuming

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}}

Apply exponent rule:

(\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}}}

Apply exponent rule:

(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}}}

3^{x} \cdot 4^{x}

2 + 2 2

What is \sqrt[4]{((7a^5b^{-6}-c)/(16b^3c))^5} ?
The solution to \sqrt[4]{((7a^5b^{-6}-c)/(16b^3c))^5} is ((7a^5-b^6c)^{5/4})/(32b^{45/4)c^{5/4}}