For this case we must find an expression equivalent to:
[tex]\sqrt [3] {\frac {75a ^ 7 * b ^ 4} {40a ^ {13} * c ^ 9}}[/tex]
We rewrite the expression as:
[tex]\sqrt [3] {\frac {5 * (15a ^ 7 * b ^ 4)} {5 (8a ^ {13} * c ^ 9)}} =\\\sqrt [3] {\frac {15a ^ 7 * b ^ 4} {8a ^ {13} * c ^ 9}} =[/tex]
By definition of power properties we have to:
[tex]\sqrt [n] {a ^ m} = a ^ {\frac {m} {n}}\\\frac {a ^ m} {a ^ n} = a ^ {m-n}[/tex]
Now, we rewrite the expression:
[tex]\sqrt [3] {\frac {15 * b ^ 4} {8a ^ 6 * c ^ 9}} =\\\sqrt [3] {\frac {b ^ 3 (15b)} {(2a ^ 2 * c ^ 3) ^ 3}} =\\\sqrt [3] {(\frac {b} {2a ^ 2 * c ^ 3}) ^ 3 * 15b} =[/tex]
[tex]\frac {b} {2a ^ 2 * c ^ 3} * \sqrt [3] {15b} =\\\frac {b * \sqrt [3] {15b}} {2a ^ 2 * c ^ 3}[/tex]
Answer:
[tex]\frac {b * \sqrt [3] {15b}} {2a ^ 2 * c ^ 3}[/tex]
