Вычислите: [tex]\sqrt{14+6\sqrt{5}}-\sqrt{9-4\sqrt{5}} [/tex]. Created by rrrrtttt01. algebra-ru

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August 2022 2 51 Report

Вычислите:
[tex]\sqrt{14+6\sqrt{5}}-\sqrt{9-4\sqrt{5}} [/tex]

Artem112

Verified answer

$\sqrt{14+6\sqrt{5}}-\sqrt{9-4\sqrt{5}} =
\sqrt{9+6\sqrt{5}+5}-\sqrt{4-4\sqrt{5}+5} =
\\\
=\sqrt{3^2+2\cdot 3\cdot\sqrt{5}+( \sqrt{5} )^2}-\sqrt{2^2-2\cdot2\cdot\sqrt{5}+( \sqrt{5} )^2} =
\\\
=\sqrt{(3+ \sqrt{5} )^2}-\sqrt{(2- \sqrt{5} )^2} =|3+ \sqrt{5} |-|2- \sqrt{5}|=
\\\
=3+ \sqrt{5} -(-(2- \sqrt{5}))=3+ \sqrt{5} +2- \sqrt{5}=5$

2 votes Thanks 2

Удачник66

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14+6√5=9+5+6√5=3^2+2*3*√5+5=(3+√5)^2
9-4√5=2^2+5-2*2*√5=(√5-2)^2
В итоге 3+√5-(√5-2)=3+2=5

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