Question

cac ban giai giup mk bai nay nha

\(\sqrt{19+8\sqrt{3}}-\sqrt{19-8\sqrt{3}}=?\)

\(\sqrt{x+2\sqrt{x-1}}+\sqrt{x-2\sqrt{x-1}}=?\)

Answer 1

\(\sqrt{19+8\sqrt{3}}-\sqrt{19-8\sqrt{3}}\)

\(=\sqrt{4^2+8\sqrt{3}+\left(\sqrt{3}\right)^2}-\sqrt{4^2-8\sqrt{3}+\left(\sqrt{3}\right)^2}\)

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

\(=\left|\sqrt{3}+4\right|-\left|\sqrt{3}-4\right|\)

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

\(=8\)

\(\sqrt{x+2\sqrt{x-1}}+\sqrt{x-2\sqrt{x-1}}\)

\(=\sqrt{\left(\sqrt{x-1}\right)^2+2\sqrt{x-1}+1^2}+\sqrt{\left(\sqrt{x-1}\right)^2-2\sqrt{x-1}+1^2}\)

\(=\sqrt{\left(\sqrt{x-1}+1\right)^2}+\sqrt{\left(\sqrt{x-1}-1\right)^2}\)

\(=\left|\sqrt{x-1}+1\right|+\left|\sqrt{x-1}-1\right|\)