←Vrati se na zadatke1. ZadatakIzračunajte 1−A−1\sqrt{1-A^{-1}}1−A−1 ako je A=2022+12022−1−22022+1.A=\frac{\sqrt{2022}+1}{\sqrt{2022}-1}-\frac{2}{\sqrt{2022}+1}.A=2022−12022+1−2022+12.RješenjeSvođenjem na zajednički nazivnik izračunat ćemo vrijednost izraza AAA. A=2022+12022−1−22022+1=(2022+1)2−2(2022−1)(2022−1)(2022+1)=2022+22022+1−22022+22022−1=20252021.\begin{aligned} A&=\frac{\sqrt{2022}+1}{\sqrt{2022}-1}-\frac{2}{\sqrt{2022}+1}\\ &=\frac{(\sqrt{2022}+1)^2-2(\sqrt{2022}-1)}{(\sqrt{2022}-1)(\sqrt{2022}+1)}\\ &=\frac{2022+2\sqrt{2022}+1-2\sqrt{2022}+2}{2022-1}\\ &=\frac{2025}{2021}. \end{aligned}A=2022−12022+1−2022+12=(2022−1)(2022+1)(2022+1)2−2(2022−1)=2022−12022+22022+1−22022+2=20212025. Tada je: 1−A−1=1−20212025=42025=245.\sqrt{1-A^{-1}}=\sqrt{1-\frac{2021}{2025}}=\sqrt{\frac4{2025}}=\frac2{45}.1−A−1=1−20252021=20254=452.
