[tex](\frac{-3x^{-6}y^{-1}z^{-2}}{6x^{-2}yz^{-5}})^{-2}-\mleft(3x^3y^2z^{-2}\mright)^3\cdot\mleft(x^{-1}y^{-2}\mright)[/tex]
That is the expression given in the exercise.
The procedure is:
1. Apply the Negative exponent rule:
[tex]=(\frac{6x^{-2}yz^{-5}}{-3x^{-6}y^{-1}z^{-2}})^2-(\frac{3x^3y^2}{z^2})^3\cdot\frac{1}{xy^2}[/tex]
2. Apply the Expanded power rule:
[tex]=\frac{36x^{-4}y^2z^{(-10)}}{9x^{(-12)}y^{-2}z^{-4}}^{}-\frac{27x^9y^6}{z^6}^{}\cdot\frac{1}{xy^2}[/tex]
3. Apply the Product rule :
[tex]=\frac{36x^{-4}y^2z^{(-10)}}{9x^{(-12)}y^{-2}z^{-4}}^{}-\frac{27x^9y^6}{z^6xy^2}[/tex]
4. Apply the Quotient rule:
[tex]=\frac{4x^8y^4^{}}{^{}z^6}^{}-\frac{27x^8y^4}{z^6^{}}[/tex]
5. Combining like terms, you get:
[tex]=\frac{4x^8y^4-27x^8y^4}{^{}z^6}^{}=-\frac{23x^8y^4}{z^6}[/tex]
The answer is:
1. Negative exponent rule.
2. Expanded power rule.
3. Product rule.
4. Quotient rule.
5. Combining like terms.
