1. **State the problem:** Simplify the expression $$6^3 \times \frac{1}{9^2}$$. 2. **Recall the formulas and rules:** - Power of a number: $$a^n$$ means multiplying $$a$$ by itself $$n$$ times. - Multiplying powers: When multiplying, calculate each power separately. - Simplify fractions by canceling common factors. 3. **Calculate each power:** $$6^3 = 6 \times 6 \times 6 = 216$$ $$9^2 = 9 \times 9 = 81$$ 4. **Rewrite the expression:** $$216 \times \frac{1}{81} = \frac{216}{81}$$ 5. **Simplify the fraction:** Find the greatest common divisor (GCD) of 216 and 81. - $$216 = 2^3 \times 3^3$$ - $$81 = 3^4$$ The common factor is $$3^3 = 27$$. 6. **Divide numerator and denominator by 27:** $$\frac{\cancel{216}^{8} \times 27}{\cancel{81}^{3} \times 27} = \frac{8}{3}$$ 7. **Final answer:** $$6^3 \times \frac{1}{9^2} = \frac{8}{3}$$