Which positive real number is equal to the quarter of its cube root?

Let x be the number to find and use the information above to write the equation 

x = (1 / 4)  3 √ x 

Multiply all terms of the equation by 4 and rewrite the equation as 

4 x =  3 √ x 

Cube both sides of the equation and simplify 

(4 x)  3  = ( 3 √ x)  3

4  3  x  3  = x 

Rewrite equation with zero on the right, factor and solve 

4  3  x  3  – x = 0 

x(4  3  x  2  – 1) = 0 

Solutions: x = 0 

and 4  3  x  2  – 1 = 0 , x  2  = 1 / 64 gives : x = ~+mn~ 1 / 8 

We are looking for a positive number, hence positive solution only 

Answer: x = 1 / 8