The Identity `sqrt(a^2)=|a|`

Simplify the following expression `sqrt(a^2)`. There are two possible cases: `a>=0` or `a<0`. If `a>=0`, then `sqrt(a^2)=a`; if `a<0`, then `sqrt(a^2)=-a`. ` `

So, `sqrt(a^2)={(a if x>=0),(-a if x<0):}`.

In the same way we determine the absolute value of real numbers. So, `sqrt(a^2)=|a|`. For example, `sqrt(3^2)=|3|=3`; `sqrt((-5)^2)=|-5|=-(-5)=5`.

In general, if `n` is even number, i.e. `n=2k`, then `root(2k)(a^(2k))=|a|`.

Example. Simplify the following: `sqrt(x^2-6x+9)+sqrt(2-x)+x-3`.

We have `sqrt(x^2-6x+9)=sqrt((x-3)^2)=|x-3|`. As the given expression contains summand `sqrt(2-x)`, then `2-x>=0`, whence we find, that `x<=2`. So, `x-3<0` and therefore `|x-3|=-(x-3)=3-x`. So, `sqrt(x^2-6x+9)=3-x` and we obtain `sqrt(x^2-6x+9)+sqrt(2-x)+x-3=3-x+sqrt(2-x)+x-3=sqrt(2-x)`.