By inspection, x=0 and x=1 are solutions

Graphically, Desmos shows a solution near 0.0117.

x ≈ 0.0117154772664753... is per Wolfram Alpha

Using Newton's method, code below, and printing out the results of different starting points yields 3 results:

    0

    0.011715477266474884

    1

The total number of real solutions appears to be 3.

Curiously, a starting point greater than 1 gives a complex number where the Real part is essentially the same as 0.011715477266474884 and the Imaginary part is a tiny negative or positive number, typically:

0.011715477266475333 ± ~3.68 x 10^(-16)i

Failed attempt to solve with algebra.  

A substitution can rid us of the fractional exponents:

t = x^(1/210)

t^30 - t^42 - t^70 + t^105 = 0

t^105 - t^70 - t^42 + t^30 = 0    take out a factor of t^30

t^30 * (t^75 - t^40 - t^12 + 1) = 0

but (t-1) is also a factor, so this could be factored as:

(t-1) * t^30 * g(t) but I'm not sure it would help.

I expect g(t) to have one real solution of approx 0.011715477266474884^(1/210) (which is about 0.979) and up to 74 complex solutions, possibly the complex 74th roots of ~0.979 (speculating).  I'm not certain if each of these would have an independent complex solution in the x domain.

def f(x):

    return  x**(1/7) - x**(1/5) - x**(1/3) + x**(1/2)

def fprime(x):

    return 1/(7*x**(6/7)) - 1/(x**(4/5)) - 1/(x**(2/3)) + 1/(x**(1/2))

def newton(first_guess, y_goal=0, epsilon = 10**(-15)):

    """  set up functions f(x) and f'(x)   """

    x_guess = first_guess

    y_error = f(x_guess) - y_goal

    

    while abs(y_error) > epsilon:

        y_error = f(x_guess) - y_goal

        x_guess -=  y_error / fprime(x_guess)

    return x_guess

###########################################

print(newton(0))

print(newton(.00001))

print(newton(1))