\(\sqrt{\left(\sqrt{2}-3\right)^2}+\sqrt{72}\)
a, \(\sqrt{200}-\sqrt{32}+\sqrt{72}\)
b, \(4\sqrt{20}-3\sqrt{125}+5\sqrt{45}-15\sqrt{\dfrac{1}{5}}\)
c, \(\left(2\sqrt{8}+3\sqrt{5}-7\sqrt{2}\right)\left(72-5\sqrt{20}-2\sqrt{2}\right)\)
a) \(\sqrt{200}-\sqrt{32}+\sqrt{72}\)
\(=\sqrt{10^2\cdot2}-\sqrt{4^2\cdot2}+\sqrt{6^2\cdot2}\)
\(=10\sqrt{2}-4\sqrt{2}+6\sqrt{2}\)
\(=\left(10-4+6\right)\sqrt{2}\)
\(=12\sqrt{2}\)
b) \(4\sqrt{20}-3\sqrt{125}+5\sqrt{45}-15\sqrt{\dfrac{1}{5}}\)
\(=4\cdot2\sqrt{5}-3\cdot5\sqrt{5}+5\cdot3\sqrt{5}-3\sqrt{5}\)
\(=8\sqrt{5}-15\sqrt{5}+15\sqrt{5}-3\sqrt{5}\)
\(=\left(8-15+15-3\right)\sqrt{5}\)
\(=5\sqrt{5}\)
c) \(\left(2\sqrt{8}+3\sqrt{5}-7\sqrt{2}\right)\left(72-5\sqrt{20}-2\sqrt{2}\right)\)
\(=\left(2\cdot2\sqrt{2}+3\sqrt{5}-7\sqrt{2}\right)\left(72-5\cdot2\sqrt{5}-2\sqrt{2}\right)\)
\(=\left(3\sqrt{5}-3\sqrt{2}\right)\left(72-10\sqrt{5}-2\sqrt{2}\right)\)
\(\left(4\sqrt{8}-\sqrt{72}+5\sqrt{\dfrac{1}{2}}\right)2\sqrt{2}\)
\(\dfrac{5+\sqrt{5}}{\sqrt{5}}+\dfrac{3+\sqrt{3}}{\sqrt{3}+1}\left(\sqrt{3}+\sqrt{5}\right)\)
a: \(\left(4\sqrt{8}-\sqrt{72}+5\sqrt{\dfrac{1}{2}}\right)\cdot2\sqrt{2}\)
\(=\left(4\cdot2\sqrt{2}-6\sqrt{2}+\dfrac{5}{\sqrt{2}}\right)\cdot2\sqrt{2}\)
\(=\left(2\sqrt{2}+\dfrac{5}{\sqrt{2}}\right)\cdot2\sqrt{2}\)
\(=2\sqrt{2}\cdot2\sqrt{2}+\dfrac{5}{\sqrt{2}}\cdot2\sqrt{2}\)
\(=8+10=18\)
b: Sửa đề:\(\dfrac{5+\sqrt{5}}{\sqrt{5}}+\dfrac{3+\sqrt{3}}{\sqrt{3}+1}-\left(\sqrt{3}+\sqrt{5}\right)\)
\(=\dfrac{\sqrt{5}\left(\sqrt{5}+1\right)}{\sqrt{5}}+\dfrac{\sqrt{3}\left(\sqrt{3}+1\right)}{\sqrt{3}+1}-\left(\sqrt{3}+\sqrt{5}\right)\)
\(=\sqrt{5}+1+\sqrt{3}-\sqrt{3}-\sqrt{5}\)
=1
\(\left(4\sqrt{8}-\sqrt{72}+5\sqrt{\dfrac{1}{2}}\right)2\sqrt{2}\\ =\left(8\sqrt{2}-6\sqrt{2}+5\sqrt{\dfrac{1}{2}}\right)2\sqrt{2}\\ =8.2.\sqrt{2}.\sqrt{2}-6.2.\sqrt{2}.\sqrt{2}+5.2.\sqrt{2}.\sqrt{\dfrac{1}{2}}\\ =32-24+10\\ =18\\ \dfrac{5+\sqrt{5}}{\sqrt{5}}+\dfrac{3+\sqrt{3}}{\sqrt{3}+1}\left(\sqrt{3}+\sqrt{5}\right)\\ =\dfrac{\sqrt{5}\left(\sqrt{5}+1\right)}{\sqrt{5}}+\dfrac{\sqrt{3}\left(\sqrt{3}+1\right)}{\sqrt{3}}\left(\sqrt{3}+\sqrt{5}\right)\\ =\sqrt{5}+1+\sqrt{3}\left(\sqrt{3}+\sqrt{5}\right)\\ =\sqrt{5}+1+3+\sqrt{15}\\ =4+\sqrt{5}+\sqrt{15}\)
Rút gọn các biểu thức sau
a) 2\(\sqrt{32}\) + 3\(\sqrt{72}-7\sqrt{50}+\sqrt{2}\) b)\(\sqrt{\left(3-\sqrt{3}\right)^2}+\sqrt{\left(2-\sqrt{3}\right)^2}\) c) \(\sqrt{11+6\sqrt{2}}-3+\sqrt{2}\)
d) \(x-4+\sqrt{16-8x+x^2}\left(x>4\right)\) e) \(\dfrac{1}{a-b}\sqrt{a^4\left(a-b\right)^2}vớia< b\)
a) \(2\sqrt{32}+3\sqrt{72}-7\sqrt{50}+\sqrt{2}\)
\(=2\cdot4\sqrt{2}+3\cdot6\sqrt{2}-7\cdot5\sqrt{2}+\sqrt{2}\)
\(=8\sqrt{2}+18\sqrt{2}-35\sqrt{2}+\sqrt{2}\)
\(=-8\sqrt{2}\)
b) \(\sqrt{\left(3-\sqrt{3}\right)^2}+\sqrt{\left(2-\sqrt{3}\right)^2}\)
\(=\left|3-\sqrt{3}\right|+\left|2-\sqrt{3}\right|\)
\(=3-\sqrt{3}+\sqrt{3}-2\)
\(=1\)
c) \(\sqrt{11+6\sqrt{2}}-3+\sqrt{2}\)
\(=\sqrt{3^2+2\cdot3\cdot\sqrt{2}+\left(\sqrt{2}\right)^2}-3+\sqrt{2}\)
\(=\sqrt{\left(3+\sqrt{2}\right)^2}-3+\sqrt{2}\)
\(=3+\sqrt{2}-3+\sqrt{2}\)
\(=2\sqrt{2}\)
d) \(x-4+\sqrt{16-8x+x^2}\left(x>4\right)\)
\(=x-4+\sqrt{x^2-8x+16}\)
\(=x-4+\sqrt{\left(x-4\right)^2}\)
\(=x-4+\left|x-4\right|\)
\(=x-4+x-4\)
\(=2x-8\)
e) \(\dfrac{1}{a-b}\sqrt{a^4\left(a-b\right)^2}\left(a< b\right)\)
\(=\dfrac{1}{a-b}\sqrt{\left[a^2\left(a-b\right)\right]^2}\)
\(=\dfrac{1}{a-b}\left|a^2\left(a-b\right)\right|\)
\(=\dfrac{-a^2\left(a-b\right)}{a-b}\)
\(=-a^2\)
Tính:
\(A=\left(\sqrt{72}-3\sqrt{24}+5\sqrt{8}\right)\sqrt{2}+4\sqrt{27}\)
\(B=\dfrac{1}{\sqrt{2}-1}+\dfrac{14}{3+\sqrt{2}}\)
\(C=\dfrac{5+3\sqrt{5}}{\sqrt{5}}+\dfrac{3\sqrt{3}}{\sqrt{3}+1}-\left(\sqrt{5}+3\right)\)
\(D=\sqrt{\left(1-\sqrt{2}\right)^2}-3\sqrt{18}+4\sqrt{\dfrac{1}{2}}\)
Tính:
\(A=\sqrt{20}-2\sqrt{45}+3\sqrt{18}+\sqrt{72}\)
\(B=4\sqrt{\left(\sqrt{3}-1\right)^2}+2\sqrt{12}+4\sqrt{\dfrac{1}{2}}\)
\(C=\left(3+\dfrac{3-\sqrt{3}}{\sqrt{3}-1}\right)\left(3-\dfrac{3+\sqrt{3}}{1+\sqrt{3}}\right)\)
\(D=\dfrac{1}{2+\sqrt{3}}+\dfrac{1}{2-\sqrt{3}}\)
a) Ta có: \(A=\sqrt{20}-2\sqrt{45}+3\sqrt{18}+\sqrt{72}\)
\(=2\sqrt{5}-6\sqrt{5}+9\sqrt{2}+6\sqrt{2}\)
\(=-4\sqrt{5}+15\sqrt{2}\)
b) Ta có: \(B=4\sqrt{\left(\sqrt{3}-1\right)^2}+2\sqrt{12}+4\sqrt{\dfrac{1}{2}}\)
\(=4\left(\sqrt{3}-1\right)+2\cdot2\sqrt{3}+\dfrac{4}{\sqrt{2}}\)
\(=4\sqrt{3}-4+4\sqrt{3}+2\sqrt{2}\)
\(=8\sqrt{3}+2\sqrt{2}-4\)
c) Ta có: \(C=\left(3+\dfrac{3-\sqrt{3}}{\sqrt{3}-1}\right)\left(3-\dfrac{3+\sqrt{3}}{1+\sqrt{3}}\right)\)
\(=\left(3+\sqrt{3}\right)\left(3-\sqrt{3}\right)\)
=9-3
=6
d) Ta có: \(D=\dfrac{1}{2+\sqrt{3}}+\dfrac{1}{2-\sqrt{3}}\)
\(=2-\sqrt{3}+2+\sqrt{3}\)
=4
Tính:
\(a)D=\left(\sqrt{2}-\sqrt{3-\sqrt{5}}\right)\left(-\sqrt{2}\right)\\ b)2\sqrt{3}\left(\sqrt{27}+2\sqrt{48}\right)-\sqrt{75}\\ c)E=\left(\sqrt{10}+\sqrt{6}\right)\sqrt{8-2\sqrt{15}}\\ d)P=\sqrt{2+\sqrt{3}}+\sqrt{2-\sqrt{3}}\)
\(e)M=-3\sqrt{50}+2\sqrt{98}-7\sqrt{72}\)
Tính:
\(a)D=\left(\sqrt{2}-\sqrt{3-\sqrt{5}}\right)\left(-\sqrt{2}\right)\\ b)2\sqrt{3}\left(\sqrt{27}+2\sqrt{48}\right)-\sqrt{75}\\ c)E=\left(\sqrt{10}+\sqrt{6}\right)\sqrt{8-2\sqrt{15}}\\ d)P=\sqrt{2+\sqrt{3}}+\sqrt{2-\sqrt{3}}\)
\(e)M=-3\sqrt{50}+2\sqrt{98}-7\sqrt{72}\)
a) Ta có: \(D=\left(\sqrt{2}-\sqrt{3-\sqrt{5}}\right)\cdot\left(-\sqrt{2}\right)\)
\(=-2+\sqrt{6-2\sqrt{5}}\)
\(=-2+\sqrt{5-2\cdot\sqrt{5}\cdot1+1}\)
\(=-2+\sqrt{\left(\sqrt{5}-1\right)^2}\)
\(=-2+\left|\sqrt{5}-1\right|\)
\(=-2+\sqrt{5}-1\)(Vì \(\sqrt{5}>1\))
\(=-3+\sqrt{5}\)
b) Ta có: \(2\sqrt{3}\left(\sqrt{27}+2\sqrt{48}\right)-\sqrt{75}\)
\(=2\sqrt{81}+4\sqrt{144}-5\sqrt{3}\)
\(=18+48-5\sqrt{3}\)
\(=66-5\sqrt{3}\)
c) Ta có: \(E=\left(\sqrt{10}+\sqrt{6}\right)\sqrt{8-2\sqrt{15}}\)
\(=\sqrt{2}\left(\sqrt{5}+\sqrt{3}\right)\sqrt{5-2\cdot\sqrt{5}\cdot\sqrt{3}+3}\)
\(=\sqrt{2}\cdot\left(\sqrt{5}+\sqrt{3}\right)\cdot\sqrt{\left(\sqrt{5}-\sqrt{3}\right)^2}\)
\(=\sqrt{2}\cdot\left(\sqrt{5}+\sqrt{3}\right)\cdot\left|\sqrt{5}-\sqrt{3}\right|\)
\(=\sqrt{2}\cdot\left(\sqrt{5}+\sqrt{3}\right)\cdot\left(\sqrt{5}-\sqrt{3}\right)\)(Vì \(\sqrt{5}>\sqrt{3}\))
\(=\sqrt{2}\cdot\left(5-3\right)\)
\(=2\sqrt{2}\)
d) Ta có: \(P=\sqrt{2+\sqrt{3}}+\sqrt{2-\sqrt{3}}\)
\(=\sqrt{\frac{3}{2}+2\cdot\sqrt{\frac{3}{2}}\cdot\sqrt{\frac{1}{2}}+\frac{1}{2}}+\sqrt{\frac{3}{2}-2\cdot\sqrt{\frac{3}{2}}\cdot\sqrt{\frac{1}{2}}+\frac{1}{2}}\)
\(=\sqrt{\left(\sqrt{\frac{3}{2}}+\sqrt{\frac{1}{2}}\right)^2}+\sqrt{\left(\sqrt{\frac{3}{2}}-\sqrt{\frac{1}{2}}\right)^2}\)
\(=\left|\sqrt{\frac{3}{2}}+\sqrt{\frac{1}{2}}\right|+\left|\sqrt{\frac{3}{2}}-\sqrt{\frac{1}{2}}\right|\)
\(=\sqrt{\frac{3}{2}}+\sqrt{\frac{1}{2}}+\sqrt{\frac{3}{2}}-\sqrt{\frac{1}{2}}\)(Vì \(\sqrt{\frac{3}{2}}>\sqrt{\frac{1}{2}}>0\))
\(=2\sqrt{\frac{3}{2}}=\sqrt{4\cdot\frac{3}{2}}=\sqrt{6}\)
e) Ta có: \(M=-3\sqrt{50}+2\sqrt{98}-7\sqrt{72}\)
\(=\sqrt{2}\cdot\left(-3\cdot\sqrt{25}+2\cdot\sqrt{49}-7\cdot\sqrt{36}\right)\)
\(=\sqrt{2}\cdot\left(-15+14-42\right)\)
\(=-43\sqrt{2}\)
tính: \(\left(2\sqrt{8}+3\sqrt{5}-7\sqrt{2}\right)\left(\sqrt{72}-5\sqrt{20}-2\sqrt{2}\right)\)
Thực hiện phép tính
\(\left(2\sqrt{8}+3\sqrt{5}-7\sqrt{2}\right)\left(\sqrt{72}-5\sqrt{20}-2\sqrt{2}\right)\)