Question 747110
<pre>
{{{root(5,192t^17u^15)}}}

Write 192 = 2<sup>6</sup>·3

{{{root(5,2^6*3*t^17u^15)}}}

Write each exponent in terms of the largest
multiple of the index of the radical which
does not exceed it.

Looking at the 2<sup>6</sup>, the largest multiple of index
5 that does not exceed 6 is 5. So we write 2<sup>6</sup> as 2<sup>5+1</sup> = 2<sup>5</sup>·2<sup>1</sup>·

{{{root(5,2^(5+1)*3*t^17u^15)}}} 

Looking at the t<sup>17</sup>, the largest multiple of index
5 that does not exceed 17 is 15. So we write t<sup>17</sup> as t<sup>15+2</sup> = t<sup>15</sup>t<sup>2</sup>·

{{{root(5,2^(5+1)*3*t^(15+2)u^15)}}}

Looking at the u<sup>15</sup>, this exponent is already a multiple of index
5 so we leave it as it.

We use the identity a<sup>b+c</sup> =a<sup>b</sup>a<sup>c</sup> to
rewrite  

2<sup>5+1</sup> = 2<sup>5</sup>2<sup>1</sup> = 2<sup>5</sup>·2,

and 

t<sup>15+2</sup> = t<sup>15+2</sup> = t<sup>15</sup>t<sup>2</sup>

{{{root(5,2^5*2*3*t^15t^2u^15)}}} 

To take the factors with exponents which are multiples of 5,
out of the radical, divide their exponent by the index 5, 
and get:

{{{2*t^3*u^3*root(5,2*3*t^2)}}}

{{{2t^3u^3*root(5,6t^2)}}}

Edwin</pre>