Question 362218
<pre>

There is no solution but we can show why there is no solution.

{{{drawing(180,80,0,4,-1,1, locate(0,.7,

 (x-1)^(1/3) = sqrt(x))  )}}}

Square both sides:

{{{drawing(180,80,0,4,-1,1, locate(0,.7,

 ((x-1)^(1/3))^2 = (sqrt(x))^2)  )}}}

Simplify:

{{{drawing(180,80,0,4,-1,1, locate(0,.7,

 (x-1)^(2/3) = x ))}}}

Cube both sides:

{{{drawing(180,80,0,4,-1,1, locate(0,.7,

 ((x-1)^(2/3))^3 = (x)^3 ))}}}

Simplify:

{{{(x-1)^2 = x^3}}}

{{{x^2-2x+1=x^3}}}

{{{0=x^3-x^2+2x-1}}}

We try 1 to see if it is a solution:

1 | 1 -1  2 -1
  |<u>    1  0  2</u> 
    1  0  2  1

1 is not a solution. However, since all numbers on the 
bottom row of the synthetic division are non-negative, 
1 is an upper bound for the solutions.  So any solutions 
are therefore less than 1.  However any solution less than
1 when substituted into the original equation, which is:  

{{{drawing(180,80,0,4,-1,1, locate(0,.7,

 (x-1)^(1/3) = sqrt(x))  )}}}

will yield a negative number on the left side.  However
the right side is non-negative because the square root
radical cannot be a negative number.  Therefore there
can be no solution.

We can check that conclusion graphically, heuristically.

Below is the graph of

{{{drawing(180,80,0,4,-1,1, locate(0,.7,

 y=(x-1)^(1/3) - sqrt(x))  )}}}

From which we can see there is no solution for y=0 since

the entire graph is below the x-axis.

{{{graph(400,100,-5,15,-2.5,2.5,(x-1)^(1/3) - sqrt(x))  }}}

Edwin</pre>