1. **State the problem:** Calculate the value of the expression $$\frac{12}{39} \times \frac{65}{28} \times \frac{21}{10} - \sqrt{\frac{25}{36}}$$. 2. **Recall formulas and rules:** - Multiplication of fractions: $$\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}$$. - Square root of a fraction: $$\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$$. - Simplify fractions by canceling common factors. 3. **Calculate the product of fractions:** $$\frac{12}{39} \times \frac{65}{28} \times \frac{21}{10} = \frac{12 \times 65 \times 21}{39 \times 28 \times 10}$$ 4. **Multiply numerators and denominators:** $$= \frac{16380}{10920}$$ 5. **Simplify the fraction:** Find the greatest common divisor (GCD) of 16380 and 10920, which is 5460. 6. **Divide numerator and denominator by 5460:** $$= \frac{\cancel{16380}^{3}}{\cancel{10920}^{2}} = \frac{3}{2}$$ 7. **Calculate the square root term:** $$\sqrt{\frac{25}{36}} = \frac{\sqrt{25}}{\sqrt{36}} = \frac{5}{6}$$ 8. **Subtract the square root from the product:** $$\frac{3}{2} - \frac{5}{6}$$ 9. **Find common denominator (6):** $$= \frac{3 \times 3}{2 \times 3} - \frac{5}{6} = \frac{9}{6} - \frac{5}{6}$$ 10. **Subtract numerators:** $$= \frac{9 - 5}{6} = \frac{4}{6}$$ 11. **Simplify the fraction:** $$= \frac{\cancel{4}^{2}}{\cancel{6}^{3}} = \frac{2}{3}$$ **Final answer:** $$\frac{2}{3}$$