# 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