Question 1154462
<pre>

{{{(8x^(1/2)y^3)^(1/3)*(25xy^(1/2))^(1/2)}}}

Put in all the invisible multiplication symbols:

{{{(8*x^(1/2)*y^3)^(1/3)*(25*x*y^(1/2))^(1/2)}}}

Put in all invisible 1-exponents, to the 8, the 25 and the x:

{{{(8^1*x^(1/2)*y^3)^(1/3)*(25^1*x^1*y^(1/2))^(1/2)}}}

Remove the parentheses by distributing the exponents:

{{{matrix(2,1,"",8^(1/3)*x^(1/6)*y^1*25^(1/2)*x^(1/2)*y^(1/4))}}}

Add the exponents of the x's and of the y's:

{{{matrix(2,1,"",8^(1/3)*x^(1/6+1/2)*y^(1+1/4)*25^(1/2))}}}

{{{matrix(2,1,"",8^(1/3)*x^(1/6+3/6)*y^(4/4+1/4)*25^(1/2))}}}

{{{matrix(2,1,"",8^(1/3)*x^(4/6)*y^(5/4)*25^(1/2))}}}

{{{matrix(2,1,"",8^(1/3)*x^(2/3)*y^(5/4)*25^(1/2))}}}

Since
      {{{matrix(2,1,"",8^(1/3)=root(3,8)=2)}}}
and 
      {{{matrix(2,1,"",25^(1/2)=sqrt(25)=5)}}}
,

{{{matrix(2,1,"",2*x^(2/3)*y^(5/4)*5)}}}

{{{matrix(2,1,"",10*x^(2/3)*y^(5/4))}}}

If your teacher wanted you to stop with fractional exponents, then
the above would be the final answer.  However, if your teacher
wanted your answer to be expressed with roots, then continue:

Express the fractional exponents with the same LCD:

{{{matrix(2,1,"",10*x^(8/12)*y^(15/12))}}}

Since
      {{{matrix(2,1,"",x^(8/12)=root(12,x^8))}}}
and 
      {{{matrix(2,1,"",y^(15/12)=root(12,y^15))}}}
      
{{{matrix(2,1,"",10*root(12,x^8)*root(12,y^15))}}}

{{{matrix(2,1,"",10*root(12,x^8*y^15))}}}

Edwin</pre>