simplify-3-2-2-

Question Number 33575 by MURALI last updated on 19/Apr/18

simplify \sqrt{(3 + 2 \sqrt{2)}}

Answered by Rasheed.Sindhi last updated on 19/Apr/18

Let \sqrt{(3 + 2 \sqrt{2)}} = a + b \sqrt{2}

where a, b \in Z

{(\sqrt{(3 + 2 \sqrt{2)}})}^{2} = {(a + b \sqrt{2})}^{2}

a^{2} + {2 b}^{2} + 2 ab \sqrt{2} = 3 + 2 \sqrt{2}

a^{2} + {2 b}^{2} = 3 \land ab = 1

ab = 1 \Rightarrow b = 1 / a

a^{2} + {2 b}^{2} = 3 \Rightarrow a^{2} + \frac{2}{a^{2}} = 3

a^{4} - {3 a}^{2} + 2 = 0

(a^{2} - 1) (a^{2} - 2) = 0

a = \pm 1, a = \pm \sqrt{2} \notin Z

Hence a = \pm 1

b = 1 / (\pm 1) = \pm 1

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

Answered by MURALI last updated on 19/Apr/18

{(3 + 2 \sqrt{2})}^{2} = {(\sqrt{2})}^{2}

Commented by Rasheed.Sindhi last updated on 21/Apr/18

Perhaps you mean

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

Answered by MURALI last updated on 19/Apr/18

(3 + 2 \sqrt{2}) = {(\sqrt{2})}^{2} + 2 \sqrt{2} \times 1 + {(1)}^{2}

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

∴ \sqrt{3 + 2 \sqrt{2)}} = \sqrt{2} + 1

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