please solve this: (√(30+12(√6)))


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*Question Number 29909 by D¬ B£$T* last updated on 13/Feb/18**

$p l e a s e s o l v e t h i s : \sqrt{30 + 12 \sqrt{6}}$
Commented by rahul 19 last updated on 13/Feb/18

$\sqrt{30 + 12 \sqrt{6}} = \sqrt{30 + 2 \sqrt{216} =} (\sqrt{a} + \sqrt{b}) .$ $so a + b = 30$ $and ab = 216$ $\Rightarrow a = \frac{216}{b} .$ $on solving we get a = 18, b = 12 .$ $hence it is equal to \sqrt{18} + \sqrt{12} .$ $or we can say 3 \sqrt{2} + 2 \sqrt{3} .$
	Answered by mrW2 last updated on 13/Feb/18
	$let 30+12(√6)=(p+q(√6))^2 =p^2 +2pq(√6)+6q^2 2pq=12⇒pq=6 p^2 +6q^2 =30 ⇒p^2 +6((6/p))^2 =30 ⇒p^4 −30p^2 +216=0 ⇒p^2 =((30±(√(30^2 −4×216)))/2)=((30±6)/2)= { ((18)),((12)) :} ⇒p= { ((3(√2))),((2(√3))) :} ⇒q= { (((6/(3(√2)))=(√2))),(((6/(2(√3)))=(√3))) :} ⇒(√(30+12(√6)))=p+q(√6)=3(√2)+(√2)×(√6)=3(√2)+2(√3) or ⇒(√(30+12(√6)))=2(√3)+(√3)×(√6)=2(√3)+3(√2)=the same as abov$
	$l e t 30 + 12 \sqrt{6} = {(p + q \sqrt{6})}^{2} = p^{2} + 2 p q \sqrt{6} + 6 q^{2}$ $2 p q = 12 \Rightarrow p q = 6$ $p^{2} + 6 q^{2} = 30$ $\Rightarrow p^{2} + 6 {(\frac{6}{p})}^{2} = 30$ $\Rightarrow p^{4} - 30 p^{2} + 216 = 0$ $\Rightarrow p^{2} = \frac{30 \pm \sqrt{30^{2} - 4 \times 216}}{2} = \frac{30 \pm 6}{2} = {\begin{cases} 18 \\ 12 \end{cases}$ $\Rightarrow p = {\begin{cases} 3 \sqrt{2} \\ 2 \sqrt{3} \end{cases}$ $\Rightarrow q = {\begin{cases} \frac{6}{3 \sqrt{2}} = \sqrt{2} \\ \frac{6}{2 \sqrt{3}} = \sqrt{3} \end{cases}$ $\Rightarrow \sqrt{30 + 12 \sqrt{6}} = p + q \sqrt{6} = 3 \sqrt{2} + \sqrt{2} \times \sqrt{6} = 3 \sqrt{2} + 2 \sqrt{3}$ $o r$ $\Rightarrow \sqrt{30 + 12 \sqrt{6}} = 2 \sqrt{3} + \sqrt{3} \times \sqrt{6} = 2 \sqrt{3} + 3 \sqrt{2} = t h e s a m e a s a b o v$
	Commented by NECx last updated on 13/Feb/18

	$s o n i c e w o r k i n g s$
	Answered by ajfour last updated on 13/Feb/18

	$l e t a = \sqrt{30 + 12 \sqrt{6}}$ $\Rightarrow a^{2} = 30 + 12 \sqrt{6} . . . . . (i)$ $\frac{1}{a^{2}} = \frac{1}{30 + 12 \sqrt{6}} = \frac{30 - 12 \sqrt{6}}{36}$ $o r \frac{36}{a^{2}} = 30 - 12 \sqrt{6} . . . (i i)$ $(i) + (i i) g i v e s :$ $a^{2} + \frac{36}{a^{2}} = 60$ $\Rightarrow {(a + \frac{6}{a})}^{2} = 60 + 12$ $\Rightarrow a + \frac{6}{a} = 6 \sqrt{2}$ $o r a^{2} - 6 \sqrt{2} a + 6 = 0$ ${(a - 3 \sqrt{2})}^{2} = 12$ $a = 2 \sqrt{3} + 3 \sqrt{2} .$
	Commented by 33 last updated on 13/Feb/18

	$t o b e h o n e s t i l i k e d m r . {a j f o u r}^{'} s$ $m e t h o d t h e m o s t a s i t i s s i m p l e s t$ $o f a l l .$
	Commented by abdo imad last updated on 13/Feb/18

	$b u t t h e m e t h o d g i v e n b y s i r {m r w}_{2} i s g e n e r a l a n d g i v e$ $o f t e n t h e a n s w e r . . .$
	Commented by 33 last updated on 13/Feb/18

	$h m m m a l r i g h t$
	Commented by mrW2 last updated on 14/Feb/18
	Variety is always good! All roads lead to Rome.
	Commented by 33 last updated on 14/Feb/18

	$h a h a y e a h$
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