Question

How do you simplify `sqrt((a +2)^2)`?

Answer 1

`a+2`

Explanation:

`sqrt((a+2)^2)` means to take the square root of `(a+2)^2`, which is `a+2`.

Answer 2

`sqrt ((a+2)^2)=a+2` `" "a in [-2,-oo)`
`" "`
`sqrt((a+2)^2) = -(a+2)` `" "a in (-oo,-2)`

Explanation:

`sqrt ((a+2)^2)=abs (a+2)`
`" "`
If`" "color (blue)(a+2>=0)rArra >= -2" "`
`" "`
then
`" "`
`sqrt ((a+2)^2)=color (blue)(a+2)` `" "a in [-2,-oo)`
`" "`
If`" "color (red)(a+2 < 0 ) rArr a <-2" "`
`" "`
then
`" "`
`sqrt((a+2)^2) = color (red)(-(a+2))` `" "a in (-oo,-2)`

Answer 3

`sqrt((a+2)^2) = abs(a+2)`

Explanation:

A square root of a number `x` is a number `y` such that `y^2 = x`.

Any non-zero number `x` has two square roots, which we write as `sqrt(x)` and `-sqrt(x)`. The principal square root is `sqrt(x)`.

If `x` is positive then `sqrt(x)` is the positive square root and `-sqrt(x)` the negative one.

If `x = t^2` for some number `t` then the square roots of `x` are `t` and `-t`.

Hence we find that the square roots of `(a+2)^2` are `(a+2)` and `-(a+2)`.

Which of `(a+2)` and `-(a+2)` is the principal, non-negative one? Whichever is positive, or if zero, then they are both the same.

We can automatically choose between the two using the absolute value and write:

`sqrt((a+2)^2) = abs(a+2)`