1. **State the problem:** Simplify the expression $$\frac{\sqrt{561^2 - 459^2}}{4 \times \frac{2}{7} \times 0.15 + 4 \times \frac{2}{7} \div \frac{20}{3}} + 4\sqrt{10} \div \left(\frac{1}{3} \sqrt{40}\right)$$. 2. **Recall formulas and rules:** - Use the difference of squares: $$a^2 - b^2 = (a-b)(a+b)$$. - Simplify fractions and multiplication carefully. - Simplify square roots by factoring out perfect squares. 3. **Calculate the numerator inside the square root:** $$561^2 = 314721$$ $$459^2 = 210681$$ $$561^2 - 459^2 = 314721 - 210681 = 104040$$ 4. **Simplify the square root:** $$\sqrt{104040}$$ Factor 104040: $$104040 = 10404 \times 10 = (102^2) \times 10$$ So, $$\sqrt{104040} = \sqrt{102^2 \times 10} = 102 \sqrt{10}$$ 5. **Simplify the denominator of the first fraction:** Calculate each part: $$4 \times \frac{2}{7} \times 0.15 = 4 \times \frac{2}{7} \times \frac{15}{100} = 4 \times \frac{2}{7} \times \frac{3}{20} = 4 \times \frac{6}{140} = 4 \times \frac{3}{70} = \frac{12}{70} = \frac{6}{35}$$ Next part: $$4 \times \frac{2}{7} \div \frac{20}{3} = 4 \times \frac{2}{7} \times \frac{3}{20} = \frac{8}{7} \times \frac{3}{20} = \frac{24}{140} = \frac{12}{70} = \frac{6}{35}$$ Sum denominator: $$\frac{6}{35} + \frac{6}{35} = \frac{12}{35}$$ 6. **First fraction:** $$\frac{102 \sqrt{10}}{\frac{12}{35}} = 102 \sqrt{10} \times \frac{35}{12} = \frac{102 \times 35}{12} \sqrt{10} = \frac{3570}{12} \sqrt{10} = 297.5 \sqrt{10}$$ 7. **Simplify the second term:** $$4 \sqrt{10} \div \left(\frac{1}{3} \sqrt{40}\right) = 4 \sqrt{10} \times \frac{3}{\sqrt{40}} = 12 \sqrt{10} \div \sqrt{40}$$ Simplify $$\sqrt{40} = \sqrt{4 \times 10} = 2 \sqrt{10}$$ So, $$12 \sqrt{10} \div (2 \sqrt{10}) = \frac{12}{2} \times \frac{\sqrt{10}}{\sqrt{10}} = 6$$ 8. **Add both parts:** $$297.5 \sqrt{10} + 6$$ **Final answer:** $$297.5 \sqrt{10} + 6$$