Question

Tính

\(a.\sqrt{5-2\sqrt{5}}+\sqrt{5+2\sqrt{5}}\)

\(b.\sqrt{7-4\sqrt{3}}+\sqrt{7+4\sqrt{3}}\)

\(c.\sqrt{13-4\sqrt{3}}+\sqrt{13+4\sqrt{3}}\)

Answer 1

Lời giải:

a)
\((\sqrt{5-2\sqrt{5}}+\sqrt{5+2\sqrt{5}})^2=5-2\sqrt{5}+5+2\sqrt{5}+2\sqrt{(5-2\sqrt{5})(5+2\sqrt{5})}\)

\(=10+2\sqrt{5^2-(2\sqrt{5})^2}=10+2\sqrt{5}\)

\(\Rightarrow \sqrt{5-2\sqrt{5}}+\sqrt{5+2\sqrt{5}}=\sqrt{10+2\sqrt{5}}\)

b)

\(\sqrt{7-4\sqrt{3}}+\sqrt{7+4\sqrt{3}}=\sqrt{2^2+3-2.2\sqrt{3}}+\sqrt{2^2+3+2.2\sqrt{3}}\)

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

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

c)

\(\sqrt{13-4\sqrt{3}}+\sqrt{13+4\sqrt{3}}=\sqrt{13-2\sqrt{12}}+\sqrt{13+2\sqrt{12}}\)

\(=\sqrt{12+1-2\sqrt{12}}+\sqrt{12+1+2\sqrt{12}}=\sqrt{(\sqrt{12}-1)^2}+\sqrt{(\sqrt{12}+1)^2}\)

\(=\sqrt{12}-1+\sqrt{12}+1=2\sqrt{12}=4\sqrt{3}\)

Answer 2

Lời giải:

a)
\((\sqrt{5-2\sqrt{5}}+\sqrt{5+2\sqrt{5}})^2=5-2\sqrt{5}+5+2\sqrt{5}+2\sqrt{(5-2\sqrt{5})(5+2\sqrt{5})}\)

\(=10+2\sqrt{5^2-(2\sqrt{5})^2}=10+2\sqrt{5}\)

\(\Rightarrow \sqrt{5-2\sqrt{5}}+\sqrt{5+2\sqrt{5}}=\sqrt{10+2\sqrt{5}}\)

b)

\(\sqrt{7-4\sqrt{3}}+\sqrt{7+4\sqrt{3}}=\sqrt{2^2+3-2.2\sqrt{3}}+\sqrt{2^2+3+2.2\sqrt{3}}\)

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

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

c)

\(\sqrt{13-4\sqrt{3}}+\sqrt{13+4\sqrt{3}}=\sqrt{13-2\sqrt{12}}+\sqrt{13+2\sqrt{12}}\)

\(=\sqrt{12+1-2\sqrt{12}}+\sqrt{12+1+2\sqrt{12}}=\sqrt{(\sqrt{12}-1)^2}+\sqrt{(\sqrt{12}+1)^2}\)

\(=\sqrt{12}-1+\sqrt{12}+1=2\sqrt{12}=4\sqrt{3}\)