If `((x^(-1)y^2)/(x^3y^(-2)))^1.\ ((x^6\ y^(-3))/(x^2\ y^3))^(1/2)=x^a y^b ,` prove that `a+b=-1` , where `x\ a n d\ y` are different positive primes.

If ((x^(-1)y^(2))/(x^(3)y^(-2)))^(1)+((x^(6)y^(-3))/(x^(2)y^(3)))^((1)/(2))=x^(a)y^(b) prove that a+b=-1, where x and y are different positive primes.

(x + 2) (x + 1) = (x -2) (x -3) (y+3)(y+2)=(y-1)(y-2)

if (x_(1),x_(2))^(2)+(y_(1)-y_(2))^(2)=a^(2), (x_(2)-x_(3))^(2)+(y_(2)-y_(3))^(2)=b^(2) (x_(3)-x_(1))^(2)+(y_(3)-y_(1))^(2)=c^(2). where a,b,c are positive then prove that 4 |{:(x_(1),,y_(1),,1),(x_(2) ,,y_(2),,1),( x_(3),, y_(3),,1):}| = (a+b+c) (b+c-a) (c+a-b)(a+b-c)

if (x_(1)-x_(2))^(2)+(y_(1)-y_(2))^(2)=a^(2), (x_(2)-x_(3))^(2)+(y_(2)-y_(3))^(2)=b^(2) (x_(3)-x_(1))^(2)+(y_(3)-y_(1))^(2)=c^(2). where a,b,c are positive then prove that 4 |{:(x_(1),,y_(1),,1),(x_(2) ,,y_(2),,1),( x_(3),, y_(3),,1):}| = (a+b+c) (b+c-a) (c+a-b)(a+b-c)

if (x_(1),x_(2))^(2)+(y_(1)-y_(2))^(2)=a^(2), (x_(2)-x_(3))^(2)+(y_(2)-y_(3))^(2)=b^(2) (x_(3)-x_(1))^(2)+(y_(3)-y_(1))^(2)=c^(2). where a,b,c are positive then prove that 4 |{:(x_(1),,y_(1),,1),(x_(2) ,,y_(2),,1),( x_(3),, y_(3),,1):}| = (a+b+c) (b+c-a) (c+a-b)(a+b-c)

If |[x^3+1, x^2, x] , [y^3+1, y^2, y] , [z^3+1, z^2, z]|=0 and x, y, z are all different then prove that xyz=-1

If x^2=y^3 , then prove that (x/y)^(3/2)+(y/x)^-(2/3)=x^(1/2)+y^(1/3) .

Factorize : (1) (x+y)(2x+3y)-(x+y)(x+1) (2) (x+y)(2a+b)-(3x-2y)(2a+b)

If a and b are different positive primes such that ((a^(-1)b^(2))/(a^(2)b^(-4)))^(7)-:((a^(3)b^(-5))/(a^(-2)b^(3)))=a^(x)b^(y), find x and y(a+b)^(-1)(a^(-1)+b^(-1))=a^(x)b^(y) find x+y+2

If y=x+x^2+x^3+...oo where|x|<1,prove that x=y/(1+y)