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Question Number 29909 by *D¬ B£$T* last updated on 13/Feb/18
please solve this:  (√(30+12(√6)))
$${please}\:{solve}\:{this}:\:\:\sqrt{\mathrm{30}+\mathrm{12}\sqrt{\mathrm{6}}} \\ $$
Commented by rahul 19 last updated on 13/Feb/18
(√(30+12(√(6 ))))= (√(30+2(√(216  ))=))((√a) +(√b)).  so a+b=30  and ab=216  ⇒a=((216)/b).  on solving we get a=18, b=12.  hence it is equal to (√(18)) + (√(12)) .  or we can say 3(√2) + 2(√3) .
$$\sqrt{\mathrm{30}+\mathrm{12}\sqrt{\mathrm{6}\:}}=\:\sqrt{\mathrm{30}+\mathrm{2}\sqrt{\mathrm{216}\:\:}=}\left(\sqrt{\mathrm{a}}\:+\sqrt{\mathrm{b}}\right). \\ $$$$\mathrm{so}\:\mathrm{a}+\mathrm{b}=\mathrm{30} \\ $$$$\mathrm{and}\:\mathrm{ab}=\mathrm{216} \\ $$$$\Rightarrow\mathrm{a}=\frac{\mathrm{216}}{\mathrm{b}}. \\ $$$$\mathrm{on}\:\mathrm{solving}\:\mathrm{we}\:\mathrm{get}\:\mathrm{a}=\mathrm{18},\:\mathrm{b}=\mathrm{12}. \\ $$$$\mathrm{hence}\:\mathrm{it}\:\mathrm{is}\:\mathrm{equal}\:\mathrm{to}\:\sqrt{\mathrm{18}}\:+\:\sqrt{\mathrm{12}}\:. \\ $$$$\mathrm{or}\:\mathrm{we}\:\mathrm{can}\:\mathrm{say}\:\mathrm{3}\sqrt{\mathrm{2}}\:+\:\mathrm{2}\sqrt{\mathrm{3}}\:. \\ $$
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
$${let}\:\mathrm{30}+\mathrm{12}\sqrt{\mathrm{6}}=\left({p}+{q}\sqrt{\mathrm{6}}\right)^{\mathrm{2}} ={p}^{\mathrm{2}} +\mathrm{2}{pq}\sqrt{\mathrm{6}}+\mathrm{6}{q}^{\mathrm{2}} \\ $$$$\mathrm{2}{pq}=\mathrm{12}\Rightarrow{pq}=\mathrm{6} \\ $$$${p}^{\mathrm{2}} +\mathrm{6}{q}^{\mathrm{2}} =\mathrm{30} \\ $$$$\Rightarrow{p}^{\mathrm{2}} +\mathrm{6}\left(\frac{\mathrm{6}}{{p}}\right)^{\mathrm{2}} =\mathrm{30} \\ $$$$\Rightarrow{p}^{\mathrm{4}} −\mathrm{30}{p}^{\mathrm{2}} +\mathrm{216}=\mathrm{0} \\ $$$$\Rightarrow{p}^{\mathrm{2}} =\frac{\mathrm{30}\pm\sqrt{\mathrm{30}^{\mathrm{2}} −\mathrm{4}×\mathrm{216}}}{\mathrm{2}}=\frac{\mathrm{30}\pm\mathrm{6}}{\mathrm{2}}=\begin{cases}{\mathrm{18}}\\{\mathrm{12}}\end{cases} \\ $$$$\Rightarrow{p}=\begin{cases}{\mathrm{3}\sqrt{\mathrm{2}}}\\{\mathrm{2}\sqrt{\mathrm{3}}}\end{cases} \\ $$$$\Rightarrow{q}=\begin{cases}{\frac{\mathrm{6}}{\mathrm{3}\sqrt{\mathrm{2}}}=\sqrt{\mathrm{2}}}\\{\frac{\mathrm{6}}{\mathrm{2}\sqrt{\mathrm{3}}}=\sqrt{\mathrm{3}}}\end{cases} \\ $$$$\Rightarrow\sqrt{\mathrm{30}+\mathrm{12}\sqrt{\mathrm{6}}}={p}+{q}\sqrt{\mathrm{6}}=\mathrm{3}\sqrt{\mathrm{2}}+\sqrt{\mathrm{2}}×\sqrt{\mathrm{6}}=\mathrm{3}\sqrt{\mathrm{2}}+\mathrm{2}\sqrt{\mathrm{3}} \\ $$$${or} \\ $$$$\Rightarrow\sqrt{\mathrm{30}+\mathrm{12}\sqrt{\mathrm{6}}}=\mathrm{2}\sqrt{\mathrm{3}}+\sqrt{\mathrm{3}}×\sqrt{\mathrm{6}}=\mathrm{2}\sqrt{\mathrm{3}}+\mathrm{3}\sqrt{\mathrm{2}}={the}\:{same}\:{as}\:{abov} \\ $$
Commented by NECx last updated on 13/Feb/18
so nice workings
$${so}\:{nice}\:{workings} \\ $$
Answered by ajfour last updated on 13/Feb/18
let  a=(√(30+12(√6)))  ⇒  a^2 =30+12(√6)   .....(i)          (1/a^2 )=(1/(30+12(√6))) = ((30−12(√6))/(36))  or    ((36)/a^2 ) = 30−12(√6)        ...(ii)  (i)+(ii) gives:         a^2 +((36)/a^2 ) = 60  ⇒    (a+(6/a))^2 =60+12   ⇒      a+(6/a) =6(√2)  or       a^2 −6(√2)a+6=0            (a−3(√2))^2 = 12         a = 2(√3)+3(√2)  .
$${let}\:\:{a}=\sqrt{\mathrm{30}+\mathrm{12}\sqrt{\mathrm{6}}} \\ $$$$\Rightarrow\:\:{a}^{\mathrm{2}} =\mathrm{30}+\mathrm{12}\sqrt{\mathrm{6}}\:\:\:…..\left({i}\right) \\ $$$$\:\:\:\:\:\:\:\:\frac{\mathrm{1}}{{a}^{\mathrm{2}} }=\frac{\mathrm{1}}{\mathrm{30}+\mathrm{12}\sqrt{\mathrm{6}}}\:=\:\frac{\mathrm{30}−\mathrm{12}\sqrt{\mathrm{6}}}{\mathrm{36}} \\ $$$${or}\:\:\:\:\frac{\mathrm{36}}{{a}^{\mathrm{2}} }\:=\:\mathrm{30}−\mathrm{12}\sqrt{\mathrm{6}}\:\:\:\:\:\:\:\:…\left({ii}\right) \\ $$$$\left({i}\right)+\left({ii}\right)\:{gives}: \\ $$$$\:\:\:\:\:\:\:{a}^{\mathrm{2}} +\frac{\mathrm{36}}{{a}^{\mathrm{2}} }\:=\:\mathrm{60} \\ $$$$\Rightarrow\:\:\:\:\left({a}+\frac{\mathrm{6}}{{a}}\right)^{\mathrm{2}} =\mathrm{60}+\mathrm{12}\: \\ $$$$\Rightarrow\:\:\:\:\:\:{a}+\frac{\mathrm{6}}{{a}}\:=\mathrm{6}\sqrt{\mathrm{2}} \\ $$$${or}\:\:\:\:\:\:\:{a}^{\mathrm{2}} −\mathrm{6}\sqrt{\mathrm{2}}{a}+\mathrm{6}=\mathrm{0} \\ $$$$\:\:\:\:\:\:\:\:\:\:\left({a}−\mathrm{3}\sqrt{\mathrm{2}}\right)^{\mathrm{2}} =\:\mathrm{12} \\ $$$$\:\:\:\:\:\:\:\boldsymbol{{a}}\:=\:\mathrm{2}\sqrt{\mathrm{3}}+\mathrm{3}\sqrt{\mathrm{2}}\:\:. \\ $$
Commented by 33 last updated on 13/Feb/18
to be honest i liked mr.ajfour′s  method the most as it is simplest  of all.
$${to}\:{be}\:{honest}\:{i}\:{liked}\:{mr}.{ajfour}'{s} \\ $$$${method}\:{the}\:{most}\:{as}\:{it}\:{is}\:{simplest} \\ $$$${of}\:{all}. \\ $$
Commented by abdo imad last updated on 13/Feb/18
but the method given by sir mrw_2  is general and give  often the answer...
$${but}\:{the}\:{method}\:{given}\:{by}\:{sir}\:{mrw}_{\mathrm{2}} \:{is}\:{general}\:{and}\:{give} \\ $$$${often}\:{the}\:{answer}… \\ $$
Commented by 33 last updated on 13/Feb/18
hmmm alright
$${hmmm}\:{alright} \\ $$
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
haha yeah
$${haha}\:{yeah} \\ $$

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