Question

Tính

Answer 1

a: \(\left(\sqrt{27}-3\sqrt2+2\sqrt6\right):3\sqrt3\)

\(=\left(3\sqrt3-3\sqrt2+2\sqrt6\right):3\sqrt3=1-\sqrt{\frac23}+\frac{2\sqrt6}{3\sqrt3}=1-\frac{\sqrt6}{3}+\frac{2\sqrt2}{3}=\frac{3-\sqrt6+2\sqrt2}{3}\)

b: \(\sqrt{\left(\sqrt3+1\right)^2}+\sqrt{\left(1-\sqrt3\right)^2}\)

\(=\sqrt3+1+\sqrt3-1=2\sqrt3\)

c: \(\left(\sqrt2+1\right)^2+\left(\sqrt2-1\right)^2\)

\(=3+2\sqrt2+3-2\sqrt2\)

=6

d: \(\left(\sqrt3+1\right)^2+\left(1-\sqrt3\right)^2\)

\(=4+2\sqrt3+4-2\sqrt3\)

=8

e: \(\sqrt{\left(\sqrt2+1\right)^2}-\sqrt{\left(1-\sqrt2\right)^2}\)

\(=\sqrt2+1-\left(\sqrt2-1\right)\)

\(=\sqrt2+1-\sqrt2+1=2\)

f: \(\sqrt{7+4\sqrt3}+\sqrt{7-4\sqrt3}\)

\(=\sqrt{\left(2+\sqrt3\right)^2}+\sqrt{\left(2-\sqrt3\right)^2}\)

\(=2+\sqrt3+2-\sqrt3=4\)

g: \(\sqrt{6+2\sqrt5}+\sqrt{6-2\sqrt5}\)

\(=\sqrt{\left(\sqrt5+1\right)^2}+\sqrt{\left(\sqrt5-1\right)^2}\)

\(=\sqrt5+1+\sqrt5-1=2\sqrt5\)

h: \(\sqrt{4-\sqrt7}-\sqrt{4+\sqrt7}\)

\(=\frac{1}{\sqrt2}\left(\sqrt{8-2\sqrt7}-\sqrt{8+2\sqrt7}\right)\)

\(=\frac{1}{\sqrt2}\left(\sqrt{\left(\sqrt7-1\right)^2}-\sqrt{\left(\sqrt7+1\right)^2}\right)\)

\(=\frac{1}{\sqrt2}\left(\sqrt7-1-\sqrt7-1\right)=-\frac{2}{\sqrt2}=-\sqrt2\)

i: \(\left(3-\sqrt5\right)\left(\sqrt{10}-\sqrt2\right)\cdot\sqrt{3+\sqrt5}\)

\(=\left(3-\sqrt5\right)\left(\sqrt5-1\right)\cdot\sqrt{6+2\sqrt5}\)

\(=\left(3-\sqrt5\right)\left(\sqrt5-1\right)\cdot\sqrt{\left(\sqrt5+1\right)^2}\)

\(=\left(3-\sqrt5\right)\left(\sqrt5-1\right)\cdot\left(\sqrt5+1\right)=\left(3-\sqrt5\right)\left(5-1\right)=4\left(3-\sqrt5\right)\)

j: \(\sqrt{9-4\sqrt5}-\sqrt{9+4\sqrt5}\)

\(=\sqrt{\left(\sqrt5-2\right)^2}-\sqrt{\left(\sqrt5+2\right)^2}\)

\(=\sqrt5-2-\sqrt5-2=-4\)

k: \(\sqrt{4+2\sqrt3}-\sqrt{4+2\sqrt3}=0\)

l: \(\left(4+\sqrt{15}\right)\left(\sqrt{10}-\sqrt6\right)\cdot\sqrt{4-\sqrt{15}}\)

\(=\left(4+\sqrt{15}\right)\left(\sqrt5-\sqrt3\right)\cdot\sqrt{8-2\sqrt{15}}\)

\(=\left(4+\sqrt{15}\right)\left(\sqrt5-\sqrt3\right)\left(\sqrt5-\sqrt3\right)=\left(4+\sqrt{15}\right)\left(8-2\sqrt{15}\right)=2\left(4+\sqrt{15}\right)\left(4-\sqrt{15}\right)\)

=2(16-15)

=2