a, \(\sqrt{23-8\sqrt{7}}\)
\(=\sqrt{16-8\sqrt{7}+7}\)
\(=\sqrt{4^2-2.4.\sqrt{7}+\left(\sqrt{7}\right)^2}\)
\(=\sqrt{\left(4-\sqrt{7}\right)^2}\)
\(=\left|4-\sqrt{7}\right|\)
\(=4-\sqrt{7}\)
b, \(\sqrt{18-8\sqrt{2}}\)
\(=\sqrt{16-8\sqrt{2}+2}\)
\(=\sqrt{4^2-2.4.\sqrt{2}+\left(\sqrt{2}\right)^2}\)
\(=\sqrt{\left(4-\sqrt{2}\right)^2}\)
\(=\left|4-\sqrt{2}\right|\)
\(=4-\sqrt{2}\)
c, \(\sqrt{12+6\sqrt{3}}\)
\(=\sqrt{9+6\sqrt{3}+3}\)
\(=\sqrt{3^2+2.3.\sqrt{3}+\left(\sqrt{3}\right)^2}\)
\(=\sqrt{\left(3+\sqrt{3}\right)^2}\)
\(=\left|3+\sqrt{3}\right|\)
\(=3+\sqrt{3}\)
d, \(\sqrt{17+12\sqrt{2}}\)
\(=\sqrt{9+12\sqrt{2}+8}\)
\(=\sqrt{3^2+3.2.2\sqrt{2}+\left(2\sqrt{2}\right)^2}\)
\(=\sqrt{\left(3+2\sqrt{2}\right)^2}\)
\(=\left|3+2\sqrt{2}\right|\)
\(=3+2\sqrt{2}\)
a) \(\sqrt{23-8\sqrt{7}}=4-\sqrt{7}\)
b) \(\sqrt{18-8\sqrt{2}}=4-\sqrt{2}\)
c) \(\sqrt{12+6\sqrt{3}}=3+\sqrt{3}\)
d) \(\sqrt{17+12\sqrt{2}}=3+2\sqrt{2}\)
