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Ҳисоб кунед:

\(\sqrt{\underbrace{44...4}_{2n}+\underbrace{11...1}_{n+1}-\underbrace{66...6}_{n}}\)

\(\sqrt{\underbrace{44...4}_{2n}+\underbrace{11...1}_{n+1}-\underbrace{66...6}_{n}}=\)

\(=\sqrt{\frac{4}{9}\cdot\underbrace{99...9}_{2n}+\frac{1}{9}\cdot\underbrace{99...9}_{n+1}-\frac{6}{9}\cdot\underbrace{99...9}_{n}}=\)

\(=\frac{1}{3}\cdot\sqrt{4\cdot(10^{2n}-1)+(10^{n+1}-1)-6\cdot(10^n-1)}=\)

\(=\frac{1}{3}\cdot\sqrt{4\cdot10^{2n}-4+10^{n+1}-1-6\cdot10^n+6}=\)

\(=\frac{1}{3}\cdot\sqrt{4\cdot10^{2n}+10^n\cdot(10-6)+1}=\)

\(=\frac{1}{3}\cdot\sqrt{4\cdot10^{2n}+4\cdot10^n+1}=\)

\(=\frac{1}{3}\cdot\sqrt{(2\cdot10^n+1)^2}=\)

\(=\frac{1}{3}\cdot(2\cdot10^n+1)=\)

\(=\frac{1}{3}\cdot2\underbrace{00...0}_{n-1}1=\)

\(=\underbrace{66...6}_{n-1}7\)

\(\sqrt{\underbrace{44...4}_{2n}+\underbrace{11...1}_{n+1}-\underbrace{66...6}_{n}}=\underbrace{66...6}_{n-1}7\)

Ҷавоб:\(\underbrace{66...6}_{n-1}7\)