Question

Rút gọn biểu thức.

a) \(\sqrt{\left(1-\sqrt{2}\right)^2}-\sqrt{3+2\sqrt{2}}+\sqrt{\left(-2\right)^6}\)

b) \(\sqrt{7+4\sqrt{3}}+\sqrt{13-4\sqrt{3}}\)

Answer 1

a) \(\sqrt{\left(1-\sqrt{x}\right)^2}-\sqrt{3+2\sqrt{2}}+\sqrt{\left(-2\right)^6}\)

\(=\sqrt{2}-1-\sqrt{\left(\sqrt{2}+1\right)^2}+\sqrt{2^6}\)

\(=\sqrt{2}-1-\sqrt{2}-1+8\)

\(=6\)

b) \(\sqrt{7+4\sqrt{3}}+\sqrt{13-4\sqrt{3}}\)

\(=\sqrt{\left(\sqrt{3}+4\right)^2}+\sqrt{\left(2\sqrt{3}-1\right)^2}\)

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

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

\(\)

Answer 2

a) \(\sqrt{\left(1-\sqrt{2}\right)^2}-\sqrt{\left(\sqrt{2}+1\right)^2}+\sqrt{\left(-2\right)^6}\)

\(=\left|1-\sqrt{2}\right|-\left|\sqrt{2}+1\right|+\left|\left(-2\right)^3\right|\)

\(=\sqrt{2}-1-\sqrt{2}-1+8=6\)

b) \(\sqrt{\left(2+\sqrt{3}\right)^2}+\sqrt{\left(2\sqrt{3}-1\right)^2}\)

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