1. The problem is to simplify the nested radical expression $$\sqrt{75 - 3\sqrt{27 + \sqrt{49}}}$$. 2. Start with the innermost radical: \(\sqrt{49} = 7\). 3. Substitute back: \(\sqrt{27 + 7} = \sqrt{34}\). 4. Now the expression is \(\sqrt{75 - 3\sqrt{34}}\). 5. To simplify \(\sqrt{75 - 3\sqrt{34}}\), consider the expression as \(\sqrt{a} - \sqrt{b}\) or of the form \(\sqrt{x - y}\), but here it does not simplify straightforwardly using simple integers. 6. Alternatively, we check if \(\sqrt{75 - 3\sqrt{34}} = \sqrt{m} - \sqrt{n}\) where \(m,n\) are positive numbers. 7. Assuming \(\sqrt{75 - 3\sqrt{34}} = \sqrt{m} - \sqrt{n}\), then by squaring both sides: $$ 75 - 3\sqrt{34} = m + n - 2\sqrt{mn} $$ 8. Equate the rational and irrational parts: - Rational part: \(75 = m + n\) - Irrational part: \(-3\sqrt{34} = -2\sqrt{mn} \implies 3\sqrt{34} = 2\sqrt{mn}\) 9. Square the irrational equation: $$ (3\sqrt{34})^2 = (2\sqrt{mn})^2 \ 9 \times 34 = 4mn \ 306 = 4mn \ mn = \frac{306}{4} = 76.5 $$ 10. We need to find \(m,n\) such that: $$ m + n = 75 \ mn = 76.5 $$ 11. The quadratic equation for \(x\) (representing either \(m\) or \(n\)) is: $$ x^2 - 75x + 76.5 = 0 $$ 12. Calculate the discriminant: $$ \Delta = 75^2 - 4 \times 1 \times 76.5 = 5625 - 306 = 5319 $$ 13. The roots are: $$ x = \frac{75 \pm \sqrt{5319}}{2} $$ 14. Since \(\sqrt{5319} \approx 72.95\), $$ x_1 = \frac{75 + 72.95}{2} = 73.975, \quad x_2 = \frac{75 - 72.95}{2} = 1.025 $$ 15. So \(m \approx 73.975\) and \(n \approx 1.025\). 16. Then: $$ \sqrt{75 - 3\sqrt{34}} = \sqrt{73.975} - \sqrt{1.025} \approx 8.6 - 1.01 = 7.59 $$ 17. This suggests no simpler radical form, so the simplified exact form is \(\sqrt{75 - 3\sqrt{27 + \sqrt{49}}} = \sqrt{75 - 3\sqrt{34}}\). 18. The numerical approximate value is \(7.59\). Final answer: $$\sqrt{75 - 3\sqrt{27 + \sqrt{49}}} = \sqrt{75 - 3\sqrt{34}} \approx 7.59$$