1. **State the problem:** Simplify the expression $$\sqrt{3 + \sqrt{7}} - \sqrt{|1 - \sqrt{7}|}$$. 2. **Recall important rules:** - The square root function $$\sqrt{x}$$ returns the non-negative root. - The absolute value $$|x|$$ is always non-negative. 3. **Evaluate the inner absolute value:** Calculate $$|1 - \sqrt{7}|$$. Since $$\sqrt{7} \approx 2.6457$$, $$1 - \sqrt{7} \approx 1 - 2.6457 = -1.6457$$. The absolute value is $$|1 - \sqrt{7}| = 1.6457$$. 4. **Rewrite the expression:** $$\sqrt{3 + \sqrt{7}} - \sqrt{1.6457}$$. 5. **Try to simplify $$\sqrt{3 + \sqrt{7}}$$:** Assume $$\sqrt{3 + \sqrt{7}} = \sqrt{a} + \sqrt{b}$$ for some positive $$a$$ and $$b$$. Then, $$3 + \sqrt{7} = (\sqrt{a} + \sqrt{b})^2 = a + b + 2\sqrt{ab}$$. Matching terms: $$a + b = 3$$ and $$2\sqrt{ab} = \sqrt{7}$$. From the second equation: $$2\sqrt{ab} = \sqrt{7} \implies \sqrt{ab} = \frac{\sqrt{7}}{2} \implies ab = \frac{7}{4}$$. 6. **Solve the system:** $$a + b = 3$$ $$ab = \frac{7}{4}$$ 7. **Find $$a$$ and $$b$$:** Solve quadratic equation for $$a$$: $$a(3 - a) = \frac{7}{4} \implies 3a - a^2 = \frac{7}{4} \implies a^2 - 3a + \frac{7}{4} = 0$$. Calculate discriminant: $$\Delta = 9 - 7 = 2$$. 8. **Roots:** $$a = \frac{3 \pm \sqrt{2}}{2}$$. Choose $$a = \frac{3 + \sqrt{2}}{2}$$ and $$b = 3 - a = \frac{3 - \sqrt{2}}{2}$$. 9. **Therefore:** $$\sqrt{3 + \sqrt{7}} = \sqrt{\frac{3 + \sqrt{2}}{2}} + \sqrt{\frac{3 - \sqrt{2}}{2}}$$. 10. **Approximate $$\sqrt{1.6457}$$:** $$\sqrt{1.6457} \approx 1.283$$. 11. **Approximate the entire expression:** Calculate $$\sqrt{3 + \sqrt{7}} \approx \sqrt{3 + 2.6457} = \sqrt{5.6457} \approx 2.376$$. 12. **Final value:** $$2.376 - 1.283 = 1.093$$. **Answer:** $$\sqrt{3 + \sqrt{7}} - \sqrt{|1 - \sqrt{7}|} \approx 1.093$$.