1. **Problem:** Calculate the value of the expression $$\frac{\sqrt{2}+1}{\sqrt{3}+1} + \frac{\sqrt{2}-1}{\sqrt{3}-1}$$ 2. **Formula and rules:** To simplify expressions with square roots in denominators, multiply numerator and denominator by the conjugate of the denominator to rationalize it. 3. **Step 1:** Rationalize the first fraction: $$\frac{\sqrt{2}+1}{\sqrt{3}+1} \times \frac{\sqrt{3}-1}{\sqrt{3}-1} = \frac{(\sqrt{2}+1)(\sqrt{3}-1)}{(\sqrt{3}+1)(\sqrt{3}-1)}$$ 4. **Step 2:** Simplify the denominator using difference of squares: $$ (\sqrt{3}+1)(\sqrt{3}-1) = (\sqrt{3})^2 - 1^2 = 3 - 1 = 2 $$ 5. **Step 3:** Expand the numerator: $$ (\sqrt{2}+1)(\sqrt{3}-1) = \sqrt{2}\sqrt{3} - \sqrt{2} + \sqrt{3} - 1 = \sqrt{6} - \sqrt{2} + \sqrt{3} - 1 $$ 6. **Step 4:** So the first fraction is: $$ \frac{\sqrt{6} - \sqrt{2} + \sqrt{3} - 1}{2} $$ 7. **Step 5:** Rationalize the second fraction: $$ \frac{\sqrt{2}-1}{\sqrt{3}-1} \times \frac{\sqrt{3}+1}{\sqrt{3}+1} = \frac{(\sqrt{2}-1)(\sqrt{3}+1)}{(\sqrt{3}-1)(\sqrt{3}+1)} $$ 8. **Step 6:** Simplify the denominator (same as before): $$ (\sqrt{3}-1)(\sqrt{3}+1) = 3 - 1 = 2 $$ 9. **Step 7:** Expand the numerator: $$ (\sqrt{2}-1)(\sqrt{3}+1) = \sqrt{2}\sqrt{3} + \sqrt{2} - \sqrt{3} - 1 = \sqrt{6} + \sqrt{2} - \sqrt{3} - 1 $$ 10. **Step 8:** So the second fraction is: $$ \frac{\sqrt{6} + \sqrt{2} - \sqrt{3} - 1}{2} $$ 11. **Step 9:** Add the two fractions: $$ \frac{\sqrt{6} - \sqrt{2} + \sqrt{3} - 1}{2} + \frac{\sqrt{6} + \sqrt{2} - \sqrt{3} - 1}{2} = \frac{(\sqrt{6} - \sqrt{2} + \sqrt{3} - 1) + (\sqrt{6} + \sqrt{2} - \sqrt{3} - 1)}{2} $$ 12. **Step 10:** Combine like terms in the numerator: $$ \sqrt{6} + \sqrt{6} = 2\sqrt{6} $$ $$ -\sqrt{2} + \sqrt{2} = 0 $$ $$ \sqrt{3} - \sqrt{3} = 0 $$ $$ -1 - 1 = -2 $$ So numerator becomes: $$ 2\sqrt{6} - 2 $$ 13. **Step 11:** Factor out 2: $$ 2(\sqrt{6} - 1) $$ 14. **Step 12:** Divide by denominator 2: $$ \frac{2(\sqrt{6} - 1)}{2} = \sqrt{6} - 1 $$ 15. **Step 13:** Approximate to check if it matches any options: $$ \sqrt{6} \approx 2.449, \quad 2.449 - 1 = 1.449 $$ Check options: A) $2\sqrt{2} \approx 2.828$ B) $\sqrt{3} + \sqrt{2} \approx 1.732 + 1.414 = 3.146$ C) $\sqrt{3} - \sqrt{2} \approx 1.732 - 1.414 = 0.318$ D) $2\sqrt{3} \approx 3.464$ E) $\sqrt{3} \approx 1.732$ None exactly matches $\sqrt{6} - 1$, so re-check the problem or options. 16. **Step 14:** Re-examining the problem, the original expression is: $$ \frac{\sqrt{2}+1}{\sqrt{3}+1} + \frac{\sqrt{2}-1}{\sqrt{3}-1} $$ The problem states the expression as two separate fractions added. 17. **Step 15:** The final simplified form is $\sqrt{6} - 1$ which is not among the options. 18. **Step 16:** Check if the problem expects the sum or the product. Since it is sum, answer is $\sqrt{6} - 1$. **Final answer:** $\boxed{\sqrt{6} - 1}$ Since this is not among the options, the closest is none. Possibly a typo in options. --- "slug": "sqrt expression", "subject": "algebra", "desmos": {"latex": "y=\frac{\sqrt{2}+1}{\sqrt{3}+1}+\frac{\sqrt{2}-1}{\sqrt{3}-1}", "features": {"intercepts": true, "extrema": true}}, "q_count": 5