x 2/3-16x 1/3 +64=0
if you let u = x^1/3, you can express the polynomial as
u^2 - 16u + 64 = 0
(u-8)^2 = 0
u = 8
So, since x = u^3,
x = 512
To solve the equation x^(2/3) - 16x^(1/3) + 64 = 0, we can use a substitution method to simplify the equation.
Let's make a substitution: let y = x^(1/3). This means that y^2 = (x^(1/3))^2 = x^(2/3). Now, we can rewrite the equation with the substitution:
y^2 - 16y + 64 = 0
This is a quadratic equation in terms of y. We can solve it using factoring or the quadratic formula. In this case, the equation can be factored:
(y - 8)(y - 8) = 0
This simplifies to:
(y - 8)^2 = 0
Now, we can solve for y by taking the square root of both sides:
y - 8 = 0
y = 8
Since we originally substituted y = x^(1/3), we substitute back to find x:
x^(1/3) = 8
Raise both sides of the equation to the power of 3 to isolate x:
(x^(1/3))^3 = 8^3
x = 512
So the solution to the equation x^(2/3) - 16x^(1/3) + 64 = 0 is x = 512.