1. **State the problem:** Simplify the complex fraction $$\frac{\frac{6a^3}{56}}{\frac{21a}{10b^2}}$$. 2. **Recall the formula:** Dividing by a fraction is the same as multiplying by its reciprocal: $$\frac{\frac{A}{B}}{\frac{C}{D}} = \frac{A}{B} \times \frac{D}{C}$$ 3. **Apply the formula:** $$\frac{6a^3}{56} \div \frac{21a}{10b^2} = \frac{6a^3}{56} \times \frac{10b^2}{21a}$$ 4. **Multiply numerators and denominators:** $$= \frac{6a^3 \times 10b^2}{56 \times 21a} = \frac{60a^3b^2}{1176a}$$ 5. **Simplify the fraction:** Cancel common factors in numerator and denominator: $$= \frac{\cancel{60}a^{3}b^{2}}{\cancel{1176}a} = \frac{60a^{3}b^{2}}{1176a}$$ Actually, let's factor and cancel step by step: - Factor numbers: $60 = 2^2 \times 3 \times 5$, $1176 = 2^3 \times 3 \times 7^2$ - Cancel $2^2$ and $3$: $$\frac{60}{1176} = \frac{2^2 \times 3 \times 5}{2^3 \times 3 \times 7^2} = \frac{5}{2 \times 7^2} = \frac{5}{98}$$ - Cancel $a$ powers: $a^{3} / a = a^{2}$ 6. **Final simplified expression:** $$\frac{5a^{2}b^{2}}{98}$$ **Answer:** $$\boxed{\frac{5a^{2}b^{2}}{98}}$$