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 

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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

3 x 3 = x 

Rewrite equation with zero on the right, factor and solve 

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 

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