Câu 1:Rút gọn biểu thức
a,\(\sqrt{2}\left(\sqrt{8}-\sqrt{32}+3\sqrt{18}\right)\)
b,\(\sqrt{72}-3\sqrt{20}-5\sqrt{2}+\sqrt{180}\)
c,\(\frac{4}{3+\sqrt{5}}-\frac{8}{1+\sqrt{5}}+\frac{15}{\sqrt{5}}\)
d,\(\sqrt{3}.\left(\frac{3}{4}\sqrt{48}-\frac{2}{5}\sqrt{75}+2\sqrt{3}\right)\)
e,\(\frac{1}{x+\sqrt{x}}+\frac{2\sqrt{x}}{x-1}-\frac{1}{x-\sqrt{x}}\)
g,\(\left(2-\sqrt{3}\right)\sqrt{26+15\sqrt{3}}-\left(2+\sqrt{3}\right)\sqrt{26-15\sqrt{3}}\)
h,\(\left(2-\sqrt{2}\right)5\sqrt{2}-2\sqrt{50}\)
i,\(\left(2\sqrt{3}+\sqrt{5}\right)\sqrt{3}\sqrt{60}\)
a, \(\sqrt{2}\left(\sqrt{8}-\sqrt{32}+3\sqrt{18}\right)=\sqrt{2}\left(2\sqrt{2}-4\sqrt{2}+9\sqrt{2}\right)=\sqrt{2}.7\sqrt{2}=14\)
b, \(\sqrt{72}-3\sqrt{20}-5\sqrt{2}+\sqrt{180}=6\sqrt{2}-6\sqrt{5}-5\sqrt{2}+6\sqrt{5}=\sqrt{2}\)
c, \(\frac{4}{3+\sqrt{5}}-\frac{8}{1+\sqrt{5}}+\frac{15}{\sqrt{5}}=\frac{4\left(3-\sqrt{5}\right)}{9-5}-\frac{8\left(\sqrt{5}-1\right)}{5-1}+3\sqrt{5}\)
\(=3-\sqrt{5}-2\left(\sqrt{5}-1\right)+3\sqrt{5}=5\)
d, \(\sqrt{3}\left(\frac{3}{4}\sqrt{48}-\frac{2}{5}\sqrt{75}+2\sqrt{3}\right)=\sqrt{3}\left(\frac{3}{4}.4\sqrt{3}-\frac{2}{5}.5\sqrt{3}+2\sqrt{3}\right)\)
\(=\sqrt{3}\left(3\sqrt{3}-2\sqrt{3}+2\sqrt{3}\right)=9\)
e, ĐK: \(x>0;x\ne1\)
\(\frac{1}{x+\sqrt{x}}+\frac{2\sqrt{x}}{x-1}-\frac{1}{x-\sqrt{x}}=\frac{1}{\sqrt{x}\left(\sqrt{x}+1\right)}+\frac{2\sqrt{x}}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}-\frac{1}{\sqrt{x}\left(\sqrt{x}-1\right)}\)
\(=\frac{\sqrt{x}-1}{\sqrt{x}\left(\sqrt{x}+1\right)\left(\sqrt{x}-1\right)}+\frac{2x}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}-\frac{\sqrt{x}+1}{\sqrt{x}\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}\)
\(=\frac{2x-2}{\sqrt{x}\left(\sqrt{x}+1\right)\left(\sqrt{x}-1\right)}=\frac{2\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}{\sqrt{x}\left(\sqrt{x}+1\right)\left(\sqrt{x}-1\right)}=\frac{2}{\sqrt{x}}\)
g, Tham khảo:
Câu hỏi của Nguyễn Công Phượng Jmg - Toán lớp 9 | Học trực tuyến
h, \(\left(2-\sqrt{2}\right)5\sqrt{2}-2\sqrt{50}=\left(2-\sqrt{2}\right)5\sqrt{2}-10\sqrt{2}\)
\(=5\sqrt{2}\left(2-\sqrt{2}-2\right)=-10\)
i, \(\left(2\sqrt{3}+\sqrt{5}\right)\sqrt{3}\sqrt{60}=\left(2\sqrt{3}+\sqrt{5}\right).\sqrt{3}.2\sqrt{15}\)
\(=\left(2\sqrt{3}+\sqrt{5}\right).6\sqrt{5}=12\text{}\sqrt{15}+30\)
