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Question Number 53736 by Mikael_Marshall last updated on 25/Jan/19

\sqrt{32 + 10 \sqrt{7}} + \sqrt{32 - 10 \sqrt{7}} = ?

Commented by Kunal12588 last updated on 25/Jan/19

j u s t w a n t t o a d d s o m e t h i n g

\sqrt{m + \sqrt{n}} g e t t i n g s q u a r e r o o t o f a s u r d

\sqrt{m + \sqrt{n}} = \sqrt{a} + \sqrt{b}

m + \sqrt{n} = a + b + 2 \sqrt{a b}

a + b = m (1)

4 a b = n

a^{2} + b^{2} + 2 a b - 4 a b = m^{2} - n

{(a - b)}^{2} = m^{2} - n

a - b = \sqrt{m^{2} - n} (2)

f r o m (1) a n d (2)

a = \frac{m + \sqrt{m^{2} - n}}{2}, b = \frac{m - \sqrt{m^{2} - n}}{2}

∴ \sqrt{m + \sqrt{n}} = \sqrt{\frac{m + \sqrt{m^{2} - n}}{2}} + \sqrt{\frac{m - \sqrt{m^{2} - n}}{2}}

g i v e n q u e s t i o n

\sqrt{32 + 10 \sqrt{7}} = \sqrt{\frac{32 + \sqrt{1024 - 700}}{2}} + \sqrt{\frac{32 - \sqrt{1024 - 700}}{2}}

= \sqrt{\frac{32 + 18}{2}} + \sqrt{\frac{32 - 18}{2}} = \sqrt{25} + \sqrt{7} = 5 + \sqrt{7}

s i m i l a r l y \sqrt{32 - 10 \sqrt{7}} = 5 - \sqrt{7}

Commented by maxmathsup by imad last updated on 25/Jan/19

w e h a v e 32 + 10 \sqrt{7} = 32 + 2.5 \sqrt{7} = 5^{2} + {(\sqrt{7})}^{2} + 2.5 \sqrt{7} = {(5 + \sqrt{7})}^{2}

a n d 32 - 10 \sqrt{7} = {(5 - \sqrt{7})}^{2} \Rightarrow

A = \sqrt{{(5 + \sqrt{7})}^{2}} + \sqrt{{(5 - \sqrt{7})}^{2}} =∣ 5 + \sqrt{7} ∣ + ∣ 5 - \sqrt{7} ∣ = 5 + \sqrt{7} + 5 - \sqrt{7} = 10 .

Commented by maxmathsup by imad last updated on 26/Jan/19

y o u a r e w e l c o m e .

Commented by Mikael_Marshall last updated on 26/Jan/19

t h a n k y o u

Answered by tanmay.chaudhury50@gmail.com last updated on 25/Jan/19

32 + 10 \sqrt{7}

= 25 + 7 + 10 \sqrt{7}

= {(5 + \sqrt{7})}^{2}

32 - 10 \sqrt{7}

= {(5 - \sqrt{7})}^{2}

s o a n s i s

\sqrt{{(5 + \sqrt{7})}^{2}} + \sqrt{{(5 - \sqrt{7})}^{2}}

= 5 + \sqrt{7} + 5 - \sqrt{7}

= 10

Commented by Mikael_Marshall last updated on 26/Jan/19

t h a n k s s o m u c h

Commented by tanmay.chaudhury50@gmail.com last updated on 26/Jan/19

m o s t w e l c o m e \dots

Answered by ajfour last updated on 25/Jan/19

x = \sqrt{p + 10 \sqrt{q}} + \sqrt{p - 10 \sqrt{q}}

x^{2} = 2 p + 2 \sqrt{p^{2} - 100 q}

x^{2} = 64 + 2 \sqrt{1024 - 700}

x = \sqrt{64 + 36} = 10 .

Commented by Mikael_Marshall last updated on 26/Jan/19

t h a n k y o u s o m u c h .