1. **Problem statement:** Calculate $$\left(\frac{4}{27}\right)^3 \div \left(\frac{3}{8}\right)^{-3}$$. 2. **Recall the rules:** - When dividing powers, $$a^m \div a^n = a^{m-n}$$ if bases are the same. - Negative exponents mean reciprocal: $$a^{-n} = \frac{1}{a^n}$$. - Power of a fraction: $$\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$$. 3. **Rewrite the expression:** $$\left(\frac{4}{27}\right)^3 \div \left(\frac{3}{8}\right)^{-3} = \left(\frac{4}{27}\right)^3 \times \left(\frac{3}{8}\right)^3$$ 4. **Calculate each power:** $$\left(\frac{4}{27}\right)^3 = \frac{4^3}{27^3} = \frac{64}{19683}$$ $$\left(\frac{3}{8}\right)^3 = \frac{3^3}{8^3} = \frac{27}{512}$$ 5. **Multiply the two fractions:** $$\frac{64}{19683} \times \frac{27}{512} = \frac{64 \times 27}{19683 \times 512}$$ 6. **Simplify numerator and denominator:** - Numerator: $$64 \times 27 = 1728$$ - Denominator: $$19683 \times 512$$ (keep as is for now) 7. **Factor to simplify:** - Note $$1728 = 12^3$$ - Note $$19683 = 27^3$$ - Note $$512 = 8^3$$ So denominator: $$19683 \times 512 = 27^3 \times 8^3 = (27 \times 8)^3 = 216^3$$ 8. **Rewrite fraction:** $$\frac{12^3}{216^3} = \left(\frac{12}{216}\right)^3 = \left(\frac{1}{18}\right)^3 = \frac{1}{18^3}$$ 9. **Calculate final value:** $$18^3 = 18 \times 18 \times 18 = 5832$$ **Final answer:** $$\boxed{\frac{1}{5832}}$$