\(1.\sqrt{3-2\sqrt{3}+1}\\ =\sqrt{\left(\sqrt{3}\right)^2-2\sqrt{3}\cdot1+1^2}\\ =\sqrt{\left(\sqrt{3}-1\right)^2}\\ =\sqrt{3}-1\\ 2.\sqrt{5-2\sqrt{5}+1}\\ =\sqrt{\left(\sqrt{5}\right)^2-2\cdot\sqrt{5}\cdot1+1^2}\\ =\sqrt{\left(\sqrt{5}-1\right)^2}\\ =\sqrt{5}-1\\ 3.\sqrt{1-2\sqrt{2}+2}\\ =\sqrt{2-2\sqrt{2}+1}\\ =\sqrt{\left(\sqrt{2}\right)^2-2\cdot\sqrt{2}\cdot1+1^2}\\ =\sqrt{\left(\sqrt{2}-1\right)^2}\\ =\sqrt{2}-1\\ 4.\sqrt{4-4\sqrt{5}+5}\\ =\sqrt{2^2-2\cdot2\cdot\sqrt{5}+\left(\sqrt{5}\right)^2}\\ =\sqrt{\left(2-\sqrt{5}\right)^2}\\ =\left|2-\sqrt{5}\right|\\ =\sqrt{5}-2\\ 5.\sqrt{4+4\sqrt{3}+3}\\ =\sqrt{2^2+2\cdot2\cdot\sqrt{3}+\left(\sqrt{3}\right)^2}\\ =\sqrt{\left(2+\sqrt{3}\right)^2}\\ =2+\sqrt{3}\\ 6.\sqrt{5-6\sqrt{5}+9}\\ =\sqrt{\left(\sqrt{5}\right)^2-2\cdot\sqrt{5}\cdot3+3^2}\\ =\sqrt{\left(\sqrt{5}-3\right)^2}\\ =\left|\sqrt{5}-3\right|\\ =3-\sqrt{5}\)
7: \(\sqrt{7-4\sqrt{7}+4}\)
\(=\sqrt{\left(\sqrt{7}\right)^2-2\cdot\sqrt{7}\cdot2+2^2}\)
\(=\sqrt{\left(\sqrt{7}-2\right)^2}=\sqrt{7}-2\)
10: \(\sqrt{x^2+8x+16}=\sqrt{x^2+2\cdot x\cdot4+4^2}\)
\(=\sqrt{\left(x+4\right)^2}=\left|x+4\right|\)
9: \(\sqrt{25-10\sqrt{5}+5}=\sqrt{5^2-2\cdot5\cdot\sqrt{5}+\left(\sqrt{5}\right)^2}\)
\(=\sqrt{\left(5-\sqrt{5}\right)^2}=\left|5-\sqrt{5}\right|=5-\sqrt{5}\)
8: \(\sqrt{6-8\sqrt{6}+16}\)
\(=\sqrt{16-2\cdot4\cdot\sqrt{6}+\left(\sqrt{6}\right)^2}\)
\(=\sqrt{\left(4-\sqrt{6}\right)^2}=\left|4-\sqrt{6}\right|=4-\sqrt{6}\)
15: \(\sqrt{4x^2+4x+1}=\sqrt{\left(2x+1\right)^2}=\left|2x+1\right|\)
14: \(\sqrt{x^2-10x+25}=\sqrt{\left(x-5\right)^2}=\left|x-5\right|\)
13: \(\sqrt{x-8\sqrt{x}+16}=\sqrt{x-2\cdot\sqrt{x}\cdot4+4^2}\)
\(=\sqrt{\left(\sqrt{x}-4\right)^2}=\left|\sqrt{x}-4\right|\)
12:\(\sqrt{x^2+2x+1}=\sqrt{\left(x+1\right)^2}=\left|x+1\right|\)
11: \(\sqrt{x-4\sqrt{x}+4}=\sqrt{x-2\cdot\sqrt{x}\cdot2+2^2}=\sqrt{\left(\sqrt{x}-2\right)^2}=\left|\sqrt{x}-2\right|\)
