Question

a,\(\sqrt{27-8\sqrt{11}}\)-\(\sqrt{11}\)

b,\(\sqrt{48-16\sqrt{8}}\)-\(\sqrt{8}\)

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Answer 1

\(\sqrt{27-8\sqrt{11}}-\sqrt{11}=\sqrt{\left(4-\sqrt{11}\right)^2}-\sqrt{11}=\left|4-\sqrt{11}\right|-\sqrt{11}=4-2\sqrt{11}\)

\(\sqrt{48-16\sqrt{8}}-\sqrt{8}=\sqrt{\left(4\sqrt{2}-4\right)^2}-2\sqrt{2}=\left|4\sqrt{2}-4\right|-2\sqrt{2}=2\sqrt{2}-4\)

Answer 2

  \(\sqrt{27-8\sqrt{11}}-\sqrt{11}\\ =\sqrt{4^2-2.4.\sqrt{11}+\left(\sqrt{11}\right)^2}-\sqrt{11}\\ =\sqrt{\left(4-\sqrt{11}\right)^2}-\sqrt{11}\\ =\left|4-\sqrt{11}\right|-\sqrt{11}\\ =4-\sqrt{11}-\sqrt{11}=4-\left(-2\right)=6\)

 \(\sqrt{48-16\sqrt{8}}-\sqrt{8}\\ =\sqrt{\left(4\sqrt{2}\right)^2-2.4\sqrt{2}.4+4^2}-\sqrt{8}\\ =\sqrt{\left(4\sqrt{2}-4\right)^2}-\sqrt{8}\\ =\left|4\sqrt{2}-4\right|-\sqrt{8}\\ =4\sqrt{2}-4-\sqrt{8}=4\sqrt{2}-4-3\sqrt{2}\\ =\sqrt{2}-4\)