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sqrt (1-√3/2) = sqrt { (4-2√3)/4 } = (√3-1)/2 sqrt (2-√15/2) = sqrt { (8-2√15)/4 } = (√5-√3)/2 x = (√5-1)/2 x^3 = { 5√5-3(5)+3√5-1 }/8 x^3 = √5-2 x^6 = 9-4√5 x^12 = 81+80-72√5 x^12 = 161 - 72√ 5

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m^3/{n^3+3}=m^3/3n^3=3nm^1 (nm ➖ 3nm+1){.125+125 ➖ }/{3+3 ➖ } 42+8+=250/{6+42}+8=250/66+{8+8 ➖ }=250/{66+16}=250/82=35 17^18 7^8 7^2^3 3^4^1^1^1 3^2^2^1 3^1^2^1 32(nm➖ 3nm+2). (n)^2 ➖ (1)^2/(m)^2={n^2 ➖ 1}/m^2={n^0+n^0 ➖ n^0+n^0 ➖}/m^2={n^1+n^1}/m^2=n^2/m^2=nm^1 (nm ➖ 1nm+1).

(ab^2 ➖ 49)={ab^0+ab^0 ➖ ab+ab ➖}={ab^1+ab^1}=ab^2 (ab ➖ 2ab+2) {a^2+a^2 ➖}+{b^2+b^2 ➖}={a^4+b^4}=ab^8 ab^4^4 ab^2^2^2^2 ab^1^1^1^2 ab^1^2 (ab ➖ 2ab+1). {a+a ➖ }+{b+b ➖ }/(a)^2 ➖ (b)^2={a^2+b^2}/{a^2 ➖ b^2}=ab^4/{ab^0+ab^0 ➖ ab^0+ab^0 ➖ }=ab^4/{ab^1+ab^1}=ab^4/ab^2=ab^2 (ab ➖ 2ab+2).

x^6 (9)^2 ➖ (20)2/25 =x^6 {81 ➖ 400}/25 {x^6 *319}/25 319x^6/25=153.4x^6 3^51.4x^6 3^3^17 1^1^17^1.2^2x^2^3 1^1.1^2xx^1^3 2x^3 (x ➖ 3x+2).

(0)+(1)+(1)=2 (xyz ➖ 1xyz+1). (1)+(1)+(1)=3(xyz ➖ 1xyz+1) (xyz ➖ 2xyz+2). {1+1 ➖ }/{xy+xy ➖ }+{z+z ➖ } ➖ (1)^2+{1+1 ➖}/{yz+yz ➖}+{x+x ➖ } ➖ (1)^2+{1+1 ➖ }/{zx+zx ➖}+{y+y ➖ } ➖ (1)^2=2/{xy^2+y^2} ➖ 1+2/{yz^2+x^2} ➖ 1+2/{zx^2+y^2} ➖ 1=2/{xyy^4+ ➖ 1}2/{yzx^4 ➖ 1}+2/{zxy^4 ➖ 1}={2/xyy^3+2/yzx^3+2/zxy^3}=6/xyyyzxzxy^6=1(xyyyyzxzxy ➖ 1 xyyyzxzxy+1).

2^3 2^3+3^3 /+2^3 3^3 +2^3 1^1 1^1+1^1/+1^1 1^1+2^3 (x ➖ 3x+2)

(1x ➖ 2^3)^2^3 ➖ (1x ➖ 1)^2^3 ➖ (x ➖ 1)^2^3/(1x ➖ 2^3)^1 ➖ (1x ➖ 1)^1 ➖ (x ➖ 1)^1 ( x➖ 1^1)^1^1 ➖ (x ➖)^1^1 (x ➖)^2^1/(x ➖ 1^3) (x ➖ )^2/(x ➖ 3) (x ➖ 3x+2).

516+2^2^2^3/2^2^3^2 ➖ 2^2^2^3 52^4 1^1^1^1^1/1^1^3^1 ➖ 1^1^1^1^1 2^3 2^2^2/1^1 1^3 1^1^2/ 1^31^2/ 32/ (x ➖ 3x+2).

(14)+(14)=28 (2^2)+(2^2) (1^1)+(1^2) (1^2) (b ➖ 2a+1) (7)+(7)=14 (3^4)+(3^4) (3^2^2)+(3^2^2).(1^1^1)+(3^1^2) (3^2) (b ➖ 3a+2).

(3^4+2^3+1/1)^2^3+(3^4+2^3+1/1)2^3 (1^2^2+1^1/)1^1+(1^2^2+1^1/)^1^1 (1^1/)+(1^2/) (x ➖ 2x+1) .

8.1 2^3.1 1^1^2.1 2.1 (y ➖ 2x+1). (1)+(2)=3 (y ➖ 2x+1).

x^9/(400 ➖x^2),(800 ➖ x^3)/x^2=x^9/{x^0+x^0 ➖ x^0+x^0 ➖}+{x^0+x^0 ➖ x^0+x^0 ➖ x^0+x^0 ➖ }/x^2{ x^9/x^2+x^3/x^2}=x^12/'^2=x^6 x^2^3 (x ➖ 3x+2).

(aabb ➖2aabb+2).(x ➖ 2x+2).

(9a^2) ➖ (56)^2={81a^4 ➖3136}=3055 30^2^3^2^3p^10^2^3^2^3p^2 5^5^2^3^2^3p^2 2^3^2^31^1^1^1p^1 2^1^1^3 23 (p ➖ 2p+1).

Second answer=-7

Answer=7

nice solution

Really good problem & solution 👋👋👋👋👋👋👋

nice problem ❤

Merry christmas

Amazing !

Mistake at 10'09"

x=1, (8+27)/(12+18) = 35/30 = 7/6 duh

👍 (a+b)^3 - a^3 - b^3 = 3 ab(a+b) (a+b)^5 - a^5 - b^5 = 5 ab (a+b) (a^2+ab+b^2) (a+b)^7 - a^7 - b^7 = 7 ab(a+b) (a^2+ab+b^2)^2 let a = 2x - 3 , b = x - 2 (5/3) { (4 x^2-12 x+9)+(2 x^2- 7 x+6)+(x^2- 4 x+4) } = 65 7 x^2 - 23 x+19 = 39 7 x^2 - 23 x - 20 = 0 x = 4 , - 5/7

Please write bigger so we can see easily

Good

eliminating b √a+(14-a)^2 = 28 let √a = x x+196+x^4 - 28 x^2-28= 0 x^4 - 28 x^2+x+168 = 0 x = 3 by RRT a = 9 from second equation a+√b = 14 √b = 5 b = 25

Very nice

63/(root2-1)=63+63root2 63+63root2-8=55+63root3=(a+broot 2)^3=a^3+root2(2b^3++3ab)+6ab^2 a(a^2+6b^2)=55 b(3a+2b^2)==63 b=3 a=1 Ans=1+3root2

9p^2-56=(3p)^2-2.3p+1+6p-1-56=(3p-1)^2+6p-57 is a perfect square if and only if : 6p-57=0=>p=57/6=19/2=9.5. now check: 9×(9.5)^2-56=9×90.25-56=812.25-56=756.25=(27.5)^2. Verified found correct ans is p=9.5.

smallest) p=3

सरकस है

Tanks for watching........

If (55+63√2) is written in form of ( a+√2b)^3= =a^3 + 3(a^2)√2b + 3a(√2b)^2 +(√2b)^3 Then a^3+6a(b^2) =55 ......(1) 3(a^2)b +b^3 =63 .....(2) Eq. 1 a(a^2 +6b^2) = 55 Since a^2 + 6b^2 is positive a is also positive Then there is only one possible value of a. That is. a = 1 Then from eq. (1) We get b = 3 or - 3 So (a,b) =(1,3) , (1, -3) But only (a,b)= (1,3) fulfill the eq. (2) So 55 + 63√2 =( 1+ 3√2)^3

*2/*33..(*read as square root )

Pretty clever way of simplifying the expression. Cheers. Dr. Ajit Thakur (USA).

√y = b => y = b² a•b = y ??? .. Maybe a•b = z ?

7744=88×88........

Let a=sqrt(x) and b=sqrt(20-x) then a^2+b^2=20 And the equation becomes a^3/b+b^3/a=34 ->a^4+b^4=34*a*b ->(a^2+b^2)^2-2*(a*b)^2=34*a*b -> 400-2*(a*b)^2=34*a*b let y=a*b 400-2*y^2=34*y ->y^2+17*y-200=0 ->(y-8)(y+25)=0 ->y=8 or y=-25 a^2+b^2=20 ->(a+b)^2-2*a*b=20->(a+b)^2=36 or (a+b)^2=-30 (a+b)^2=-30 has no real solutions. Hence (a+b)^2=36 ->a+b=6 or a+b=-6 a+b=6 and a*b=8 we get a=2,b=4 or a=4,b=2 ->a=2=sqrt(x) x=4 and a=4=sqrt(4),x=16 that’s our answer. a+6=-6 and a*b=8 we get a=-2,b=-4 or a=-4,b=-2 which gives no real solutions of x.

😊

make p = + - 3 which will equate the condition of perfect square.

Sum of the factors is a multiple (NOT a factor) of 6

P=3

Amazing explanation

Jo minus minus multiply kar ke minus kaise hua,hamre hain mein to minus,minus multiply karne se positive hots thaa,aap kis hain ke hain jee.

(3p-x)^2=9p^2-6px+x^2=9p^2-56 6px-x^2-56=0 X^2-6px+56=0 6px=18x p=3

Great

P=5