How do you simplify #sqrt(y^6)#?

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Answer 1 · 2016-03-27T11:55:17

#=color(green)(y^3#

#sqrt ( y^6#

Square root , can also be called as second root, in terms of fraction, second root is a half power (#color(blue)(1/2#)

So,
#sqrt ( y^6) = y^(6 * color(blue)(1/2)#

#sqrt ( y^6) = y^3#

#=color(green)(y^3#

Answer 2 · 2016-03-27T19:42:30

Another way of writing the same thing

#y^3#

#color(blue)("Method 1")#

#y^6-= y^2xxy^2xxy^2#

So #sqrt(y^6)=sqrt(y^2xxy^2xxy^2) = yxxyxxy#

Thus #color(green)(sqrt(y^6)=y^3)#

But #y^3 cab be negative and this solution is always positive so Allan's recommendation ties this down by stating:

Thus #color(green)(sqrt(y^6)=|y^3|)#

For any value # x -> |x|# is always positive

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(blue)("Method 2")#

Suppose we had #sqrt(x)# then another way of writing this is#" "x^(1/2)#

So #sqrt(y^6) = (y^6)^(1/2)#

and #(y^6)^(1/2)" is the same as "y^(6xx1/2)" "=" "y^(6/2)#

But #6/2 =3#

So #color(green)(y^(6/3)" is the same as "y^3)#