ANSWER
[tex]\sqrt[3]{- 729{a}^{9} {b}^{6} } = - 9 {a}^{3} {b}^{2} [/tex]
EXPLANATION
We want to find the cube root of
[tex] - 729 {a}^{9} {b}^{6} [/tex]
We express this symbolically as:
[tex] \sqrt[3]{- 729 {a}^{9} {b}^{6} } [/tex]
The expression under the radical called the radicand.
We need to express this radical in exponential form using the property,
[tex] {x}^{ \frac{m}{n} } = \sqrt[n]{ {x}^{m} } [/tex]
Applying this rule gives us:
[tex]\sqrt[3]{- 729 {a}^{9} {b}^{6} } = ({- 729 {a}^{9} {b}^{6}})^{ \frac{1}{3} } [/tex]
[tex]\sqrt[3]{- 729{a}^{9} {b}^{6} } = ({- {9}^{3} {a}^{9} {b}^{6}})^{ \frac{1}{3} } [/tex]
Recall that
[tex] ({a}^{m} )^{n} = {a}^{mn} [/tex]
We apply this rule on the RHS to get,
[tex]\sqrt[3]{- 729{a}^{9} {b}^{6} } = ({- {9}^{3 \times { \frac{1}{3} } } {a}^{9 \times { \frac{1}{3} } } {b}^{6 \times { \frac{1}{3} } }})[/tex]
This simplifies to
[tex]\sqrt[3]{- 729{a}^{9} {b}^{6} } = - 9 {a}^{3} {b}^{2} [/tex]
