Question 174534
<pre><font size = 4 color = "indigo"><b>
Since square roots are never negative we must in the
end be sure that our answer cannot be negative.  But
first:

{{{sqrt(5xy^7)sqrt(10x^3y^3)}}}

Multiply what's under the radicals.

{{{sqrt((5xy^7)(10x^3y^3))}}}

Simplify

{{{sqrt(50x^4y^10)}}}

The largest perfect square that will divide 
into 50 is 25, so write {{{50}}} as {{{25*2}}}

{{{sqrt(25*2x^4y^10)}}}

We can take out the square root of 25 as 5
in front of the radical:

{{{5*sqrt(2x^4y^10)}}}

The index of a square root is understood
to be 2, like this:

{{{5*root(2,2x^4y^10)}}}

And we divide the exponents of x and y by
the index 2, getting 2 and 5.  Since there
is no remainder, we will have no x's or
y's left under the radical, but only the 2.

{{{5*x^2y^5*sqrt(2)}}}

Now we have just one more thing to take
care of.  Since we know that the answer
cannot be negative, we look at the factors:
The {{{5}}} is never negative.
The {{{x^2}}} is never negative
The {{{sqrt(2)}}} is never negative.

However the {{{y^5}}} just might be negative,
because if {{{y}}} were a negative number,
then {{{y^5}}} would also be a negative
number.  So we must put absolute value
bars around the {{{y}}} to keep it from causing
the expression to be negative, so the
final answer is 

{{{ 5 * x^2 * abs(y)^5*sqrt(2) }}}

Edwin</pre>