916/VB02. problem / Math - Solved Math Problems

MR-916 / 206f. problem

Calculate the following expression:

${(\sqrt{6 - 2 \sqrt{5}} - \sqrt{6 + 2 \sqrt{5}})}^{2}$

All Problems

Source: Vene T. Bogoslavov: Collection of Solved Problems #2 - 206.f.)

Algebra → Roots → Complex exercises

${(\sqrt{6 - 2 \sqrt{5}} - \sqrt{6 + 2 \sqrt{5}})}^{2} =$

${(\sqrt{6 - \sqrt{2^{2} \cdot 5}} - \sqrt{6 + \sqrt{2^{2} \cdot 5}})}^{2} =$

${(\sqrt{6 - \sqrt{20}} - \sqrt{6 + \sqrt{20}})}^{2} =$

Let's try to eliminate root under the root using the Lagrangian identity:

\sqrt{A \pm \sqrt{B}} = \sqrt{\frac{A + \sqrt{A^{2} - B}}{2}} \pm \sqrt{\frac{A - \sqrt{A^{2} - B}}{2}}

A ⩾ 0, B ⩾ 0, A^{2} \geq B

$A = 6, B = 20$

${(\sqrt{6 - \sqrt{20}} - \sqrt{6 + \sqrt{20}})}^{2} =$

${(\sqrt{\frac{6 + \sqrt{36 - 20}}{2}} - \sqrt{\frac{6 - \sqrt{36 - 20}}{2}} - \sqrt{\frac{6 + \sqrt{36 - 20}}{2}} - \sqrt{\frac{6 - \sqrt{36 - 20}}{2}})}^{2} =$

${(\sqrt{\frac{6 + \sqrt{16}}{2}} - \sqrt{\frac{6 - \sqrt{16}}{2}} - \sqrt{\frac{6 + \sqrt{16}}{2}} - \sqrt{\frac{6 - \sqrt{16}}{2}})}^{2} =$

${(\sqrt{\frac{6 + 4}{2}} - \sqrt{\frac{6 - 4}{2}} - \sqrt{\frac{6 + 4}{2}} - \sqrt{\frac{6 - 4}{2}})}^{2} =$

${(\sqrt{5} - \sqrt{1} - \sqrt{5} - \sqrt{1})}^{2} =$

${(- 1 - 1)}^{2} =$

${(- 2)}^{2} =$

$4$

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