Quick question regarding radicals and using absolute value

[max]

If sqrt(x^2) = l x l
What if the exponent is odd?

What if sqrt(x^3) = (x)sqrt(x)

sqrt(x^5) = (x^2)sqrt(x)

sqrt(x^7) = (x^3)sqrt(x)

Should I use absolute value bars?
I was thinking that you can't because if the variable is a negative number, it would make the radicand undefined. Would the variable have to be positive not needing the absolute value bars?

And for a problem like sqrt(x^6y^3), would (I x^3 I y)sqrt(y) be correct?

 

[mmm4444bot]

Yup, yup. The radicand must be non-negative in a square root; therefore, if the radicand is a power of x with an odd exponent, then the domain of x must be constrained to x >= 0, so no absolute value symbols are needed after simplifying the radical.

Also, |x^2| does not need absolute value symbols because it equals x^2 for all Real x.

It's the same for all |x^n|, where n is even.

We write sqrt(x^3) = x * sqrt(x), and it's defined for all x >= 0.

sqrt(x^6 * y^3) = |x^3| * y * sqrt(y) is correct, too, for x = any Real number and y >= 0.

 

[max]

Thanks!