c. sqrt (13+30sqrt (2+sqrt {9+4sqrt {2))}}

1. **Innermost square root:** $9 + 4\sqrt{2} = 8 + 1 + 4\sqrt{2} = (2\sqrt{2})^2 + 2(2\sqrt{2})(1) + 1^2 = (2\sqrt{2} + 1)^2$. Therefore, $\sqrt{9+4\sqrt{2}} = 2\sqrt{2} + 1$.<br /><br />2. **Next inner square root:** $2 + \sqrt{9+4\sqrt{2}} = 2 + 2\sqrt{2} + 1 = 3 + 2\sqrt{2} = (\sqrt{2})^2 + 2(\sqrt{2})(1) + 1^2 = (\sqrt{2} + 1)^2$. Therefore, $\sqrt{2 + \sqrt{9+4\sqrt{2}}} = \sqrt{2} + 1$.<br /><br />3. **Outer square root:** $13 + 30\sqrt{2 + \sqrt{9+4\sqrt{2}}} = 13 + 30(\sqrt{2} + 1) = 13 + 30\sqrt{2} + 30 = 43 + 30\sqrt{2} = 25 + 18 + 30\sqrt{2} = 5^2 + 2(5)(3\sqrt{2}) + (3\sqrt{2})^2 = (5 + 3\sqrt{2})^2$.<br /><br />4. **Final result:** $\sqrt{13+30\sqrt{2+\sqrt{9+4\sqrt{2}}}} = 5 + 3\sqrt{2}$.<br />