Question

\(\sqrt{10-2\sqrt{21}}-\sqrt{10+2\sqrt{21}}\)

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Answer 1

Answer 2

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Answer 3

\(\sqrt{10-2\sqrt{21}}-\sqrt{10+2\sqrt{21}}\)

\(=\sqrt{7-2\sqrt{21}+3}-\sqrt{7+2\sqrt{21}+3}\)

\(=\sqrt{\left(\sqrt{7}\right)^2-2\sqrt{21}+\left(\sqrt{3}\right)^2}-\sqrt{\left(\sqrt{7}\right)^2+2\sqrt{21}+\left(\sqrt{3}\right)^2}\)

\(=\sqrt{\left(\sqrt{7}-\sqrt{3}\right)^2}-\sqrt{\left(\sqrt{7}+\sqrt{3}\right)^2}\)

\(=\left|\sqrt{7}-\sqrt{3}\right|-\left|\sqrt{7}+\sqrt{3}\right|\)

\(=\sqrt{7}-\sqrt{3}-\left(\sqrt{7}+\sqrt{3}\right)\)

\(=\sqrt{7}-\sqrt{3}-\sqrt{7}-\sqrt{3}\)

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

 

Answer 4

\(\sqrt{10-2\sqrt{21}}-\sqrt{10+2\sqrt{21}}\)

\(=\sqrt{\left(\sqrt{7}\right)^2-2.\sqrt{7}.\sqrt{3}+\left(\sqrt{3}\right)^2}-\sqrt{\left(\sqrt{7}\right)^2+2.\sqrt{7}.\sqrt{3}+\left(\sqrt{3}\right)^2}\)

\(=\sqrt{\left(\sqrt{7}-\sqrt{3}\right)^2}-\sqrt{\left(\sqrt{7}+\sqrt{3}\right)^2}=\left|\sqrt{7}-\sqrt{3}\right|-\left|\sqrt{7}+\sqrt{3}\right|\)

\(=\sqrt{7}-\sqrt{3}-\sqrt{7}-\sqrt{3}=-2\sqrt{3}\)

Answer 5

\(\sqrt{10-2\sqrt{21}}-\sqrt{10+2\sqrt{21}}=\sqrt{7}-\sqrt{3}-\sqrt{7}-\sqrt{3}=-2\sqrt{3}\)