1. **State the problem:** Simplify the expression $$\frac{30x^{-8}y^{9}z^{7}}{6y^{-4}z^{-2}}$$ and write the answer using only positive exponents. 2. **Recall the rules:** - When dividing like bases, subtract the exponents: $$a^{m} \div a^{n} = a^{m-n}$$ - Negative exponents mean reciprocal: $$a^{-m} = \frac{1}{a^{m}}$$ - Simplify coefficients by division. 3. **Simplify the coefficients:** $$\frac{30}{6} = 5$$ 4. **Simplify each variable using exponent subtraction:** - For $x$: $$x^{-8}$$ (no $x$ in denominator, so stays as is) - For $y$: $$y^{9} \div y^{-4} = y^{9 - (-4)} = y^{13}$$ - For $z$: $$z^{7} \div z^{-2} = z^{7 - (-2)} = z^{9}$$ 5. **Rewrite the expression:** $$5x^{-8}y^{13}z^{9}$$ 6. **Convert negative exponents to positive:** $$x^{-8} = \frac{1}{x^{8}}$$ 7. **Final simplified expression:** $$\frac{5y^{13}z^{9}}{x^{8}}$$ **Answer:** $$\frac{5y^{13}z^{9}}{x^{8}}$$