Bài 1.5
a.
\(=\left(2\sqrt{2^2.5}-\sqrt{3^2.5}+3\sqrt{5}\right).\sqrt{5}\)
\(=\left(4\sqrt{5}-3\sqrt{5}+3\sqrt{5}\right).\sqrt{5}\)
\(=4\sqrt{5}.\sqrt{5}=4.5=20\)
b.
\(=\left(\sqrt{4^2.2}-\sqrt{3^2.2}+2\sqrt{2}\right).\sqrt{2}-\sqrt{5^2}\)
\(=\left(4\sqrt{2}-3\sqrt{2}+2\sqrt{2}\right).\sqrt{2}-5\)
\(=3\sqrt{2}.\sqrt{2}-5=3.2-5\)
\(=1\)
c.
\(=\left(27+1+6\sqrt{3}\right)-6\sqrt{3}+2\sqrt{3}.\sqrt{3}\)
\(=28+6\sqrt{3}-6\sqrt{3}+6\)
\(=34\)
d.
\(=\sqrt{4-2\sqrt{3}}.\left(\sqrt{3}+1\right)=\sqrt{\left(\sqrt{3}-1\right)^2}.\left(\sqrt{3}+1\right)\)
\(=\left|\sqrt{3}-1\right|.\left(\sqrt{3}+1\right)=\left(\sqrt{3}-1\right)\left(\sqrt{3}+1\right)\)
\(=3-1=2\)
1.6
b.
\(=\left(\sqrt{10}+\sqrt{2}\right)\left(3-\sqrt{5}\right).\sqrt{2}.\sqrt{3+\sqrt{5}}\)
\(=\left(\sqrt{10}+\sqrt{2}\right)\left(3-\sqrt{5}\right)\sqrt{6+2\sqrt{5}}\)
\(=\left(\sqrt{10}+\sqrt{2}\right)\left(3-\sqrt{5}\right)\sqrt{5+2\sqrt{5}+1}\)
\(=\left(\sqrt{10}+\sqrt{2}\right)\left(3-\sqrt{5}\right)\sqrt{\left(\sqrt{5}+1\right)^2}\)
\(=\sqrt{2}\left(\sqrt{5}+1\right)\left(3-\sqrt{5}\right)\left(\sqrt{5}+1\right)\)
\(=\sqrt{2}\left(\sqrt{5}+1\right)^2\left(3-\sqrt{5}\right)\)
\(=\sqrt{2}\left(6+2\sqrt{5}\right)\left(3-\sqrt{5}\right)\)
\(=2\sqrt{2}\left(3+\sqrt{5}\right)\left(3-\sqrt{5}\right)\)
\(=2\sqrt{2}.\left(9-5\right)\)
\(=8\sqrt{2}\)
1.6.
c.
\(=\sqrt{2-\sqrt{3}}.\sqrt{2}.\left(\sqrt{3}-1\right)\left(2+\sqrt{3}\right)\)
\(=\sqrt{4-2\sqrt{3}}.\left(\sqrt{3}-1\right)\left(2+\sqrt{3}\right)\)
\(=\sqrt{3-2\sqrt{3}+1}.\left(\sqrt{3}-1\right)\left(2+\sqrt{3}\right)\)
\(=\sqrt{\left(\sqrt{3}-1\right)^2}\left(\sqrt{3}-1\right)\left(2+\sqrt{3}\right)\)
\(=\left|\sqrt{3}-1\right|\left(\sqrt{3}-1\right).\left(2+\sqrt{3}\right)\)
\(=\left(\sqrt{3}-1\right)^2\left(2+\sqrt{3}\right)\)
\(=\left(4-2\sqrt{3}\right)\left(2+\sqrt{3}\right)\)
\(=2\left(2-\sqrt{3}\right)\left(2+\sqrt{3}\right)\)
\(=2.\left(4-3\right)\)
\(=2\)
f.
\(=\left(\sqrt[]{6+2.\sqrt[]{6}.\sqrt[]{5}+5}-\sqrt[]{6-2.\sqrt[]{6}.\sqrt[]{2}+2}\right)\left(\sqrt[]{5}-\sqrt[]{3}\right)\)
\(=\left(\sqrt[]{\left(\sqrt[]{6}+\sqrt[]{5}\right)^2}-\sqrt[]{\left(\sqrt[]{6}-\sqrt[]{2}\right)^2}\right)\left(\sqrt[]{5}-\sqrt[]{2}\right)\)
\(=\left(\left|\sqrt[]{6}+\sqrt[]{5}\right|-\left|\sqrt[]{6}-\sqrt[]{2}\right|\right)\left(\sqrt[]{5}-\sqrt[]{2}\right)\)
\(=\left(\sqrt[]{5}+\sqrt[]{2}\right)\left(\sqrt[]{5}-\sqrt[]{2}\right)\)
\(=5-2=3\)
Bài 1.6
a.
\(=\left(4+\sqrt{15}\right).\left(\sqrt{5}-\sqrt{3}\right).\sqrt{2}.\sqrt{4-\sqrt{15}}\)
\(=\left(4+\sqrt{15}\right)\left(\sqrt{5}-\sqrt{3}\right).\sqrt{8-2\sqrt{15}}\)
\(=\left(4+\sqrt{15}\right).\left(\sqrt{5}-\sqrt{3}\right).\sqrt{5-2.\sqrt{5}.\sqrt{3}+3}\)
\(=\left(4+\sqrt{15}\right)\left(\sqrt{5}-\sqrt{3}\right).\sqrt{\left(\sqrt{5}-\sqrt{3}\right)^2}\)
\(=\left(4+\sqrt{15}\right)\left(\sqrt{5}-\sqrt{3}\right)\left|\sqrt{5}-\sqrt{3}\right|\)
\(=\left(4+\sqrt{15}\right)\left(\sqrt{5}-\sqrt{3}\right)^2\)
\(=\left(4+\sqrt{15}\right)\left(8-2\sqrt{15}\right)\)
\(=2\left(4+\sqrt{15}\right)\left(4-\sqrt{15}\right)\)
\(=2.\left(4^2-15\right)=2\)
1.6.
d.
\(=\left(\sqrt{6}+\sqrt{2}+\sqrt{3}+1\right)\left(5-2\sqrt{2}-\sqrt{3}\right)\left(\sqrt{6}+1\right)\)
\(=\left(2\sqrt{6}-2\right)\left(\sqrt{6}+1\right)\)
\(=2\left(\sqrt{6}-1\right)\left(\sqrt{6}+1\right)\)
\(=2\left(6-1\right)=10\)
e.
\(=\dfrac{\sqrt{3+2\sqrt[]{3}+1}-\sqrt{3}}{\left(\sqrt{5}+2\right)\sqrt[3]{5\sqrt[]{5}-3.5.2+3.\sqrt[]{5}.4-8}-2}\)
\(=\dfrac{\sqrt[]{\left(\sqrt[]{3}+1\right)^2}-\sqrt[]{3}}{\left(\sqrt[]{5}+2\right)\sqrt[3]{\left(\sqrt[]{5}-2\right)^3}-2}\)
\(=\dfrac{\sqrt[]{3}+1-\sqrt[]{3}}{\left(\sqrt[]{5}+2\right)\left(\sqrt[]{5}-2\right)-2}=\dfrac{1}{5-4-2}\)
\(=-1\)
