Question

Rút gọn biểu thức chứa căn

a) \(\sqrt{9-4\sqrt{5}}\)- \(\sqrt{5}\)

b) \(\sqrt{4+2\sqrt{ }3}\)-\(\sqrt{3}\)

c) \(\sqrt{4-2\sqrt{3}}\)-\(\sqrt{3}\)

d) \(\sqrt{4+2\sqrt{3}}\)-\(\sqrt{4-2\sqrt{3}}\)

Answer 1

a. 

$\sqrt{9-4\sqrt{5}}-\sqrt{5}=\sqrt{2^2-2.2\sqrt{5}+5}-\sqrt{5}$

$=\sqrt{(2-\sqrt{5})^2}-\sqrt{5}=|2-\sqrt{5}|-\sqrt{5}=\sqrt{5}-2-\sqrt{5}=-2$

b.

$\sqrt{4+2\sqrt{3}}-\sqrt{3}=\sqrt{3+2\sqrt{3}+1}-\sqrt{3}=\sqrt{(\sqrt{3}+1)^2}-\sqrt{3}$

$=|\sqrt{3}+1|-\sqrt{3}=\sqrt{3}+1-\sqrt{3}=1$

c.

$=\sqrt{3-2\sqrt{3}+1}-\sqrt{3}=\sqrt{(\sqrt{3}-1)^2}-\sqrt{3}=|\sqrt{3}-1|-\sqrt{3}$
$=\sqrt{3}-1-\sqrt{3}=-1$

d.

$=\sqrt{(\sqrt{3}+1)^2}-\sqrt{(\sqrt{3}-1)^2}=|\sqrt{3}+1|-|\sqrt{3}-1|=(\sqrt{3}+1)-(\sqrt{3}-1)$

$=2$