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\(\sqrt{5\sqrt{3}+5\sqrt{48-10\sqrt{7+\sqrt{48}}}}\)
\(=\sqrt{5\sqrt{3}+5\sqrt{48-10\sqrt{\left(2-\sqrt{3}\right)^2}}}\)
\(=\sqrt{5\sqrt{3}+5\sqrt{48-20+10\sqrt{3}}}\)
\(=\sqrt{5\sqrt{3}+5\sqrt{\left(5-\sqrt{3}\right)^2}}\)
\(=\sqrt{5\sqrt{3}+25-5\sqrt{3}}\)
= 5
\(\dfrac{\sqrt{3}-\sqrt{5+\sqrt{24}}+\sqrt{\sqrt{72}+11}}{\sqrt{6+\sqrt{20}}+\sqrt{2}-\sqrt{7+\sqrt{40}}}\)
\(=\dfrac{\sqrt{3}-\sqrt{\left(\sqrt{2}+\sqrt{3}\right)^2}+\sqrt{\left(3+\sqrt{2}\right)^2}}{\sqrt{\left(\sqrt{5}+1\right)^2}+\sqrt{2}-\sqrt{\left(\sqrt{2}+\sqrt{5}\right)^2}}\)
\(=\dfrac{\sqrt{3}-\sqrt{2}-\sqrt{3}+3+\sqrt{2}}{\sqrt{5}+1+\sqrt{2}-\sqrt{2}-\sqrt{5}}\)
\(=3\)
Ta có: \(\sqrt{5\sqrt{3}+5\sqrt{48-10\sqrt{7+4\sqrt{3}}}}\)
\(=\sqrt{5\sqrt{3}+5\sqrt{48-10\left(2+\sqrt{3}\right)}}\)
\(=\sqrt{5\sqrt{3}+5\sqrt{28-10\sqrt{3}}}\)
\(=\sqrt{5\sqrt{3}+5\left(5-\sqrt{3}\right)}\)
\(=\sqrt{25}=5\)
\(\sqrt{5\sqrt{3+5\sqrt{48}-10\sqrt{7+4\sqrt{3}}}}\)
=\(\sqrt{5\sqrt{3+5\sqrt{48}-10\sqrt{4+2.2\sqrt{3+\left(\sqrt{3}\right)^2}}}}\)
=\(\sqrt{5\sqrt{3+5\sqrt{48-10.\sqrt{\left(2+\sqrt{3}\right)^2}}}}\)
=\(\sqrt{5\sqrt{3+5\sqrt{48-10.\sqrt{\left(2+\sqrt{3}\right)}}}}\)
=\(\sqrt{5\sqrt{3+5\sqrt{48-20+10}\sqrt{3}}}\)
=\(\sqrt{5\sqrt{3+5\sqrt{28+10}\sqrt{3}}}\)
=\(\sqrt{5\sqrt{3+5\sqrt{\left(\sqrt{3}\right)^2}+2.5.\sqrt{3}+5^2}}\)
=\(\sqrt{5\sqrt{3+5\sqrt{\left(\sqrt{3}+5\right)^2}}}\)
=\(\sqrt{5\sqrt{3+5\sqrt{\left(\sqrt{3}+5\right)}}}\)
=\(\sqrt{5\sqrt{3+5\sqrt{3+10}}}\)
=\(10\sqrt{3+10}\)
=\(\sqrt{10\left(\sqrt{3+1}\right)}\)
Bài làm:
Ta có: \(\sqrt{4+\sqrt{5\sqrt{3+5\sqrt{48-10\sqrt{7+4\sqrt{3}}}}}}\)
\(=\sqrt{4+\sqrt{5\sqrt{3+5\sqrt{48-10\sqrt{\left(2+\sqrt{3}\right)^2}}}}}\)
\(=\sqrt{4+\sqrt{5\sqrt{3+5\sqrt{48-10\left(2+\sqrt{3}\right)}}}}\)
\(=\sqrt{4+\sqrt{5\sqrt{3+5\sqrt{28-10\sqrt{3}}}}}\)
\(=\sqrt{4+\sqrt{5\sqrt{3+5\sqrt{\left(5-\sqrt{3}\right)^2}}}}\)
\(=\sqrt{4+\sqrt{5\sqrt{3+5\left(5-\sqrt{3}\right)}}}\)
\(=\sqrt{4+\sqrt{5\sqrt{28-5\sqrt{3}}}}\)
Đến đây chịu-.-
\(\sqrt{48}-2\sqrt{75}+\sqrt{108}-\frac{1}{7}\sqrt{147}\)
\(=4\sqrt{3}-10\sqrt{3}+6\sqrt{3}-\sqrt{3}\)
\(=\sqrt{3}\left(4-10+6-1\right)\)
\(=-\sqrt{3}\)
A = \(\sqrt{48}-2\sqrt{75}+\sqrt{108}-\dfrac{1}{7}\sqrt{147}\)
= \(\sqrt{16.3}-2\sqrt{25.3}+\sqrt{36.3}-\dfrac{1}{7}\sqrt{49.3}\)
= \(4\sqrt{3}-10\sqrt{3}+6\sqrt{3}-\sqrt{3}\)
= \(-\sqrt{3}\)