For this case we must simplify the following expression:
[tex]\frac {6-3 \sqrt [3] {6}} {\sqrt [3] {9}}[/tex]
Multiplying the numerator and denominator by[tex](\sqrt [3] {9}) ^ 2[/tex]
[tex]\frac {6-3 \sqrt [3] {6}} {\sqrt [3] {9}} * \frac {(\sqrt [3] {9}) ^ 2} {(\sqrt [3] { 9}) ^ 2} =[/tex]
We rewrite:
[tex]\frac {\frac {6-3 \sqrt [3] {6}} * (\sqrt [3] {9}) ^ 2} {\sqrt [3] {9} * (\sqrt [3] {9 }) ^ 2} =[/tex]
By properties of powers we have that:
[tex]a ^ m * a ^ n = a ^ {m + n}\\\frac {(6-3 \sqrt [3] {6}) * (\sqrt [3] {9}) ^ 2} {(\sqrt [3] {9}) ^ 3} =\\\frac {(6-3 \sqrt [3] {6}) * (\sqrt [3] {9}) ^ 2} {9} =[/tex]
We rewrite, moving the exponent within the radical:
[tex]\frac {(6-3 \sqrt [3] {6}) * \sqrt [3] {9 ^ 2}} {9} =\\\frac {(6-3 \sqrt [3] {6}) * \sqrt [3] {81}} {9} =[/tex]
We can rewrite[tex]3 * 3 ^ 3 = 81[/tex]
[tex]\frac {(6-3 \sqrt [3] {6}) * \sqrt [3] {3 * 3 ^ 3}} {9} =[/tex]
We simplify:
[tex]\frac {(6-3 \sqrt [3] {6}) * 3 \sqrt [3] {3}} {9} =[/tex]
We apply distributive property:
[tex]\frac {18 \sqrt [3] {3} -9 \sqrt [3] {18}} {9} =[/tex]
Simplifying we finally have:
[tex]2 \sqrt [3] {3} - \sqrt [3] {18}[/tex]
Answer:
[tex]2 \sqrt [3] {3} - \sqrt [3] {18}[/tex]
