sqrtx4

sqrtx4 refers to the expression sqrt(x^4), the square root of x to the fourth power. For real numbers x, x^4 is nonnegative, so sqrt(x^4) is well defined and nonnegative. Using the general rule sqrt(a^2) = |a| with a = x^2 gives sqrt(x^4) = |x^2|. Since x^2 is itself nonnegative for all real x, |x^2| = x^2. Therefore, for all real x, sqrt(x^4) simplifies to x^2.

In the real number system, this identity holds universally. In the complex plane, however, the square root is multi-valued and depends on branch choices, so sqrt(z^4) may correspond to z^2 or -z^2 depending on the chosen branch. Thus the straightforward simplification sqrt(x^4) = x^2 is typically stated for real x, with caution about extensions to complex numbers.

Notes:

- sqrt(x^4) = x^2 for real x.

- The expression is defined for all real x, unlike (sqrt x)^4, which is only defined when x ≥ 0 in the real numbers.

- This simplification is frequently used in algebra to reduce expressions involving even powers.