How do you simplify #\sqrt { 50x ^ { 6} y ^ { 8} } #?

We can rewrite this expression as:

#sqrt(25x^6y^8 * 2)#

Now, using this rule for radicals we can simplify the expression:

#sqrt(color(red)(a) * color(blue)(b)) = sqrt(color(red)(a)) * sqrt(color(blue)(b))#

#sqrt(color(red)(25x^6y^8) * color(blue)(2)) =>#

#sqrt(color(red)(25x^6y^8)) * sqrt(color(blue)(2)) =>#

#5x^3y^4 * sqrt(color(blue)(2)) =>#

#5x^3y^4sqrt(2)#

I start by separating the coefficient (number) and variables.

What is the square root for 50?

50 breaks up into # 25 xx 2#. You want to use 25 because it's a perfect square and easy to find the square root.

#sqrt 50 = sqrt 25 sqrt 2#

The square root of 25 is 5 so can be simplified further:
#sqrt 50 = sqrt 25 sqrt 2= 5 sqrt 2#

Now look at the variables. You should divided the exponents by 2. That number will go on the outside of the root. If there is any remainder that will go on the inside of the room.

#sqrt (x^6 y^8) #

look at #x^6# take the 6 divide it by 2...you get 3.
so it looks like

#x^3 sqrt (y^8)#

now you have the #y^8# take the 8 divide it by 2..you get 4 so it looks like

#x^3 y^4#

Now combine squish your parts together. Parts outside of the square root stay together and parts inside the square root go together.

#5x^3y^4sqrt2#

DONE!!