Question

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

Answer 1

`=color(green)(y^3`

Explanation:

`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

Another way of writing the same thing

`y^3`

Explanation:

`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)`