X^1/3 x Y^1/2=6 求 X x Y=?

(1)X^(1/3) * Y^(1/2) = 6
X^2 * Y^3 = 6^6.
(XY)^2 * Y = 6^6................(1)
Since Y = X +1, therefore,
(XY)^2(X+1) = 6^6
(XY)^2 * X + (XY)^2 = 6^6
(XY)^2 *X = 6^6 - (XY)^2.................(2)
(1) * (2) we get
(XY)^2 * Y *(XY)^2 * X = 6^6[6^6 - (XY)^2]
(XY)^5 = 6^12 - 6^6(XY)^2
(XY)^5 + 6^6(XY)^2 - 6^12 = 0
Solution to this equation will be the value of XY.
(ii)
X+ Y = 2^8
(X+Y)^2 = 2^16
X^2 + Y^2 + 2XY = 2^16..............(1)
XY= 2^12
4XY = 2^14.....................(2)
(1) - (2) we get
X^2 +Y^2 - 2XY = 2^16 - 2^14 = 2^14(4-1) = 2^14 *3
(X-Y)^2 = 2^14 * 3
X-Y = 2^7 sqrt3
X^2 - Y^2 = (X+Y)(X-Y) = 2^8 * 2^7 sqrt3 = 2^15sqrt3.

x1/3(y1/2)=6
y1/2=6x-1/3
y=36x-2/3
y=x+1
36x-2/3=x+1
x-36x-2/3+1=0
x2-36x1/3+x=0
Let f(x)=x2-36x1/3+x
f(8)=64-72+8=0
(x-8) is a root of x
When x=8, y=9
xy=72

(x-y)2=(x2-2xy+y2)
=(x2+2xy+y2-4xy)
=(x+y)2-4xy
=216-4(212)
=216-214
=214(4-1)
=3(214)
(x2-y2)=(x+y)(x-y)
=28(27sqrt3)
=215sqrt3