Question

Rút gọn biểu thức 1. \(D=\sqrt{5}-\sqrt{13-4\sqrt{9-4\sqrt{5}}}\)

2. \(B=2\sqrt{125}+\sqrt{\left(1-\sqrt{5}\right)^2}-\frac{4}{\sqrt{5}+1}\)

3.\(C=\frac{2}{\sqrt{3}+1}-\frac{1}{\sqrt{3}-2}+\frac{12}{\sqrt{3}+3}\)

Answer 1

\(D=\sqrt{5}-\sqrt{13-4\sqrt{\left(\sqrt{5}-2\right)^2}}=\sqrt{5}-\sqrt{13-4\left(\sqrt{5}-2\right)}\)

\(=\sqrt{5}-\sqrt{21-4\sqrt{5}}=\sqrt{5}-\sqrt{\left(2\sqrt{5}-1\right)^2}\)

\(=\sqrt{5}-2\sqrt{5}+1=1-\sqrt{5}\)

\(B=10\sqrt{5}+\left|1-\sqrt{5}\right|-\frac{4\left(\sqrt{5}-1\right)}{\left(\sqrt{5}+1\right)\left(\sqrt{5}-1\right)}\)

\(=10\sqrt{5}+\sqrt{5}-1-\sqrt{5}+1=10\sqrt{5}\)

\(C=\frac{2\left(\sqrt{3}-1\right)}{\left(\sqrt{3}+1\right)\left(\sqrt{3}-1\right)}+\frac{2+\sqrt{3}}{\left(2+\sqrt{3}\right)\left(2-\sqrt{3}\right)}+\frac{12\left(3-\sqrt{3}\right)}{\left(3+\sqrt{3}\right)\left(3-\sqrt{3}\right)}\)

\(=\sqrt{3}-1+2+\sqrt{3}+2\left(3-\sqrt{3}\right)=7\)