Solve `(a^x/a^y)times(a^y/a^z)times(a^z/a^x)=`

Let x,y,z be the vector, such that |x|=|y|=|z| =sqrt(2) and x,y,z make angles of 60^(@) with each other. If x times (y times z) = a . The value of y is:

Using properties of determinants, prove that |(a+x, y, z),(x, a+y, z),(x, y,a+z)|=a^2(a+x+y+z)

Show that: ((a^(x+1))/(a^(y+1)))^(x+y)\ ((a^(y+2))/(a^(z+2)))^(y+z)\ ((a^(z+3))/(a^(x+3)))^(z+x)=1

Solve the equations by matrix method: x+y+z=6 x+y-z=0 2x+y+2z=10

Prove: |(z, x, y),( z^2,x^2,y^2),(z^4,x^4,y^4)|=|(x, y, z),( x^2,y^2,z^2),(x^4,y^4,z^4)|=|(x^2,y^2,z^2),(x^4,y^4,z^4),(x, y, z)|=x y z(x-y)(y-z)(z-x)(x+y+z) .

evaluate: |(a+x,y,z),(x,a+y,z),(x,y,a+z)|

Solve the equations by matrix method : (i) x+2y+z=7 x+2y+z=7 x+3z=11

Prove that |(a-x)^2(a-y)^2(a-z)^2(b-x)^2(b-y)^2(b-z)^2(c-x)^2(c-y)^2(c-z)^2|= |(1+a x)^2(1+b x)^2(1+c x)^2(1+a y)^2(1+b y)^2(1+c y)^2(1+a z)^2(1+b z)^2(1+c z)^2|=2(b-c)(c-c)(a-b)xx(y-z)(z-x)(x-y)dot

Solve the system of equations in x,y and z satisfying the following equations x+[y]+{z}=3.1, y+[z]+{x}=4.3 and z+[x]+{y}=5.4

Prove that [(x^2+y^2+z^2)/(x+y+z)]^(x+y+z)> x^x y^y z^z >[(x+y+z)/3]^(x+y+z)(x ,y ,z >0)