explain why (-100)0.2 is possible but (-100)0.5 is not

1. Recall that a fractional exponent, like x^{\frac{1}{n}} , translates to the nth root of x.
2. For real numbers:
- (-100)^{0.2} = (-100)^{\frac{1}{5}} corresponds to finding the 5th root of -100. This operation is valid because the 5th root of a negative number is also negative, resulting in a real number.
- Therefore:
(-100)^{0.2} \approx -0.72477 + 1.37638i
3. For (-100)^{0.5} = (-100)^{\frac{1}{2}} , it requires finding the square root of -100. The square root of a negative number is not defined in the real number system and results in an imaginary number.
- Therefore:
(-100)^{0.5} \quad \text{is invalid in the real number system (complex result: approximately } \quad \pm 10i)

The 5th root of -100 is possible because it results in a real number, while the square root of -100 is not possible in the real domain since it leads to an imaginary number. Thus, the expression (-100)^{0.5} is invalid in the real number system.