a: \(\sqrt{\left(11-6\sqrt2\right)^2}+\sqrt{\left(11+6\sqrt2\right)^2}\)
\(=11-6\sqrt2+11+6\sqrt2\)
=22
b: \(\sqrt{\left(10-4\sqrt6\right)^2}-\sqrt{\left(10+4\sqrt6\right)^2}\)
\(=10-4\sqrt6-10-4\sqrt6=-8\sqrt6\)
c: \(\sqrt{\left(4-\sqrt5\right)^2}+\sqrt{\left(1-\sqrt5\right)^2}\)
\(=4-\sqrt5+\left|1-\sqrt5\right|\)
\(=4-\sqrt5+\sqrt5-1=4-1=3\)
d: \(\sqrt{\left(7+\sqrt2\right)^2}-\sqrt{\left(1-\sqrt2\right)^2}\)
\(=7+\sqrt2-\left(\sqrt2-1\right)=7+\sqrt2-\sqrt2+1\)
=7+1
=8
e: \(\sqrt{6+2\sqrt5}+\sqrt{6-2\sqrt5}\)
\(=\sqrt{\left(\sqrt5+1\right)^2}+\sqrt{\left(\sqrt5-1\right)^2}\)
\(=\sqrt5+1+\sqrt5-1=2\sqrt5\)
f: \(\sqrt{8-2\sqrt7}-\sqrt{8+2\sqrt7}\)
\(=\sqrt{\left(\sqrt7-1\right)^2}-\sqrt{\left(\sqrt7+1\right)^2}\)
\(=\sqrt7-1-\sqrt7-1=-2\)
g: \(\sqrt{11+6\sqrt2}-\sqrt{11-6\sqrt2}\)
\(=\sqrt{\left(3+\sqrt2\right)^2}-\sqrt{\left(3-\sqrt2\right)^2}\)
\(=3+\sqrt2-\left(3-\sqrt2\right)=2\sqrt2\)
h: \(\sqrt{17+12\sqrt2}+\sqrt{17-12\sqrt2}\)
\(=\sqrt{\left(3+2\sqrt2\right)^2}+\sqrt{\left(3-2\sqrt2\right)^2}\)
\(=3+2\sqrt2+3-2\sqrt2=6\)
i: Đặt \(A=\sqrt{a^2+6a+9}+\sqrt{a^2-6a+9}\)
\(=\sqrt{\left(a+3\right)^2}+\sqrt{\left(a-3\right)^2}\)
=|a+3|+|a-3|
-3<=a<=3
=>a+3>=0 và a-3<=0
A=|a+3|+|a-3|
=a+3+3-a
=6
k: 1<=a<2
=>a-1>=0 và \(\sqrt{a-1}-1<0\)
\(\sqrt{a+2\sqrt{a-1}}+\sqrt{a-2\sqrt{a-1}}\)
\(=\sqrt{a-1+2\cdot\sqrt{a-1}\cdot1+1}+\sqrt{a-1-2\cdot\sqrt{a-1}\cdot1+1}\)
\(=\sqrt{\left(\sqrt{a-1}+1\right)^2}+\sqrt{\left(\sqrt{a-1}-1\right)^2}=\sqrt{a-1}+1+1-\sqrt{a-1}=2\)
