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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

$$justwanttoaddsomething$$$$\sqrt{m+\sqrt{n}}gettingsquarerootofasurd$$$$\sqrt{m+\sqrt{n}}=\sqrt{a}+\sqrt{b}$$$$m+\sqrt{n}=a+b+2\sqrt{ab}$$$$a+b=m\left(1\right)$$$$4ab=n$$$$a^2+b^2+2ab-4ab=m^2-n$$$${(a-b)}^2=m^2-n$$$$a-b=\sqrt{m^2-n}\left(2\right)$$$$from\left(1\right)and\left(2\right)$$$$a=\frac{m+\sqrt{m^2-n}}{2},b=\frac{m-\sqrt{m^2-n}}{2}$$$$\therefore \sqrt{m+\sqrt{n}}=\sqrt{\frac{m+\sqrt{m^2-n}}{2}}+\sqrt{\frac{m-\sqrt{m^2-n}}{2}}$$$$givenquestion$$$$\sqrt{32+10\sqrt{7}}=\sqrt{\frac{32+\sqrt{1024-700}}{2}}+\sqrt{\frac{32-\sqrt{1024-700}}{2}}$$$$=\sqrt{\frac{32+18}{2}}+\sqrt{\frac{32-18}{2}}=\sqrt{25}+\sqrt{7}=5+\sqrt{7}$$$$similarly\sqrt{32-10\sqrt{7}}=5-\sqrt{7}$$

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

$$wehave32+10\sqrt{7}=32+2.5\sqrt{7}=5^2+{\left(\sqrt{7}\right)}^2+2.5\sqrt{7}={(5+\sqrt{7})}^2$$$$and32-10\sqrt{7}={(5-\sqrt{7})}^2\Rightarrow$$$$A=\sqrt{{(5+\sqrt{7})}^2}+\sqrt{{(5-\sqrt{7})}^2}=\mid 5+\sqrt{7}\mid +\mid 5-\sqrt{7}\mid =5+\sqrt{7}+5-\sqrt{7}=10.$$

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

$$youarewelcome.$$

Commented by Mikael_Marshall last updated on 26/Jan/19

$$thankyou$$

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

$$32+10\sqrt{7}$$$$=25+7+10\sqrt{7}$$$$={(5+\sqrt{7})}^2$$$$32-10\sqrt{7}$$$$={(5-\sqrt{7})}^2$$$$soansis$$$$\sqrt{{(5+\sqrt{7})}^2}+\sqrt{{(5-\sqrt{7})}^2}$$$$=5+\sqrt{7}+5-\sqrt{7}$$$$=10$$

Commented by Mikael_Marshall last updated on 26/Jan/19

$$thankssomuch$$

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

$$mostwelcome...$$

Answered by ajfour last updated on 25/Jan/19

$$x=\sqrt{p+10\sqrt{q}}+\sqrt{p-10\sqrt{q}}$$$$x^2=2p+2\sqrt{p^2-100q}$$$$x^2=64+2\sqrt{1024-700}$$$$x=\sqrt{64+36}=10.$$

Commented by Mikael_Marshall last updated on 26/Jan/19

$$thankyousomuch.$$

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