Lời giải:
Ta có:
\(\frac{\sqrt{3}+\sqrt{5}}{(\sqrt{5}+1)(\sqrt{3}-1)}=\frac{(\sqrt{3}+\sqrt{5})(\sqrt{5}-1)(\sqrt{3}+1)}{(\sqrt{5}+1)(\sqrt{5}-1)(\sqrt{3}-1)(\sqrt{3}+1)}\)
\(=\frac{(\sqrt{3}+\sqrt{5})(\sqrt{5}-1)(\sqrt{3}+1)}{(5-1)(3-1)}=\frac{(\sqrt{3}+\sqrt{5})(\sqrt{5}-1)(\sqrt{3}+1)}{8}\)
Trục căn ở mẫu:
\(a)\frac{5}{\sqrt{10}}\\ b)\frac{-2}{1-\sqrt{5}}\\ c)\frac{4}{\sqrt{3}+\sqrt{2}}\\ d)\frac{1}{3-2\sqrt{2}}\\ e)\frac{6-\sqrt{6}}{1-\sqrt{6}}\\ g)\frac{3\sqrt{2}-2\sqrt{3}}{2\left(\sqrt{3}-\sqrt{2}\right)}\\ h)\frac{\sqrt{3}-3}{\sqrt{3}-1}\\ i)\frac{\sqrt{15}}{5\sqrt{3}+3\sqrt{5}}\)
Trục căn thức ở mẫu
a)\(\dfrac{\sqrt{6}+\sqrt{14}}{2\sqrt{3}-\sqrt{7}}\)
b)\(\dfrac{5\sqrt{5}+3\sqrt{3}}{\sqrt{5}+\sqrt{3}}\)
c)\(\dfrac{3+4\sqrt{3}}{\left(\sqrt{6}+\sqrt{2}\right)-\sqrt{5}}\)
Rút gọn biểu thức:
a)\(\frac{2}{\sqrt{5}-\sqrt{3}}+\frac{3}{\sqrt{6}+\sqrt{3}}\)
b)\(\frac{3+2\sqrt{3}}{\sqrt{3}}+\frac{2+\sqrt{2}}{\sqrt{2}+1}-\left(2+\sqrt{3}\right)\)
c)\(\sqrt{3+\sqrt{5}}-\sqrt{3-\sqrt{5}}-\sqrt{2}\)
d)\(\left(1+tan^2a\right)\left(1-sin^2a\right)+\left(1+cotan^2a\right)\left(1-cos^2a\right)\)
1) thực hiện phép tính
\(3\sqrt{12}+\dfrac{1}{2}\sqrt{48}-\sqrt{27}\)
2) trục căn thức ở mẫu : \(\dfrac{2}{\sqrt{3}-5}\)
3) khử mẫu của biểu thức lấy căn: \(\sqrt{\dfrac{2}{5}}\)
Chứng minh
\(\frac{1}{3\left(1+\sqrt{2}\right)}+\frac{1}{5\left(\sqrt{2}+\sqrt{3}\right)}+\frac{1}{7\left(\sqrt{3}+\sqrt{4}\right)}+...+\frac{1}{97\left(\sqrt{48}+\sqrt{49}\right)}< \frac{3}{7}\)
Giải phương trình \(\dfrac{3\left(x-\sqrt{3}\right)\left(x-\sqrt{5}\right)}{\left(1-\sqrt{3}\right)\left(1-\sqrt{5}\right)}+\dfrac{4\left(x-1\right)\left(x-\sqrt{5}\right)}{\left(\sqrt{3}-1\right)\left(\sqrt{3}-\sqrt{5}\right)}+\dfrac{5\left(x-1\right)\left(x-\sqrt{3}\right)}{\left(\sqrt{5}-1\right)\left(\sqrt{5}-\sqrt{3}\right)}=3x-2\)
Tính
\(A=\frac{3}{\sqrt{3}}+\frac{2\sqrt{3}}{\sqrt{3}+1}\) \(B=\frac{\sqrt{15}-\sqrt{12}}{\sqrt{5}-2}-\frac{1}{2-\sqrt{3}}\)
\(C=\frac{5+2\sqrt{5}}{\sqrt{5}}+\frac{3+\sqrt{3}}{\sqrt{3}}-\left(\sqrt{5}+\sqrt{3}\right)\)
\(D=\sqrt{\frac{4}{\left(2-\sqrt{5}\right)^2}}-\sqrt{\frac{4}{\left(2+\sqrt{5}\right)^2}}\) \(E=\frac{\sqrt{10}-\sqrt{2}}{\sqrt{5}-1}-\frac{2-\sqrt{2}}{\sqrt{2}-1}\)
Gpt: a) \(\sqrt[4]{3\left(x+5\right)}-\sqrt[4]{11-x}=\sqrt[4]{13+x}-\sqrt[4]{3\left(3-x\right)}\)
b) \(\frac{1+2\sqrt{x}-x\sqrt{x}}{3-x-\sqrt{2-x}}=2\left(\frac{1+x\sqrt{x}}{1+x}\right)\) c) \(\sqrt{x+1}+\frac{4\left(\sqrt{x+1}+\sqrt{x-2}\right)}{3\left(\sqrt{x-2}+1\right)^2}=3\)
d) \(\sqrt{\frac{x-2}{x+1}}+\frac{x+2}{\left(\sqrt{x+2}+\sqrt{x-2}\right)^2}=1\) e) \(2x+1+x\sqrt{x^2+2}+\left(x+1\right)\sqrt{x^2+2x+2}=0\)
f) \(\sqrt{2x+3}\cdot\sqrt[3]{x+5}=x^2+x-6\)
3. a.\(\sqrt{\left(4-\sqrt{17}\right)^2}\)
b.\(\frac{2\sqrt{3}}{2}\)
c \(\frac{\sqrt{6}+\sqrt{14}}{\text{2√3+√28}}\)
d.\(\frac{x+1}{\sqrt{x^2-1}}\)
e.\(\frac{x^2-5}{x+\sqrt{5}}\)
f.\(\frac{2}{2-\sqrt{3}}\)
g.\(\frac{\sqrt{2}+1}{\sqrt{2}-1}\)
f.\(\frac{x\sqrt{x}-1}{\sqrt{x}-1}\)
i.\(\frac{3}{\sqrt{20}}+\frac{1}{\sqrt{60}}-2\sqrt{\frac{1}{15}}\)
k.\(\frac{3}{\sqrt{5}-\sqrt{2}}+\frac{4}{\sqrt{6}+\sqrt{2}}\)
i.(\(\frac{1}{\sqrt{5}-\sqrt{3}}+\frac{1}{\sqrt{5}+\sqrt{3}}\))\(\sqrt{5}\)
h.\(\left(\sqrt{20}-\sqrt{45}+\sqrt{5}\right)\sqrt{5}\)
l.\(\left(5\sqrt{3}+3\sqrt{5}\right):\sqrt{15}\)
m.\(\frac{1}{3}\sqrt{48}+3\sqrt{75}-\sqrt{27}-10\sqrt{\frac{4}{3}}\)
n.\(\left(5\sqrt{\frac{1}{5}}+\frac{1}{2}\sqrt{20}-\frac{5}{4}\sqrt{\frac{4}{5}+\sqrt{5}}\right):2\sqrt{5}\)
d\(\left(2+\sqrt{5}\right)^2-\left(2+\sqrt{5}\right)^2\)
