Recall that one of the rules of exponents states that
![x^{ \frac{m}{n} }= \left(\sqrt[n]{x} \right)^m](https://tex.z-dn.net/?f=x%5E%7B%20%5Cfrac%7Bm%7D%7Bn%7D%20%7D%3D%20%5Cleft%28%5Csqrt%5Bn%5D%7Bx%7D%20%5Cright%29%5Em)
Now, let x be a negative number and n, an even number, then
![(-x)^{ \frac{m}{n} }= \left(\sqrt[n]{(-x)} \right)^m](https://tex.z-dn.net/?f=%28-x%29%5E%7B%20%5Cfrac%7Bm%7D%7Bn%7D%20%7D%3D%20%5Cleft%28%5Csqrt%5Bn%5D%7B%28-x%29%7D%20%5Cright%29%5Em)
But the even root of a negative number is not a real number.
for example,
is not a real number, rather a complex number.
Hence, <span>rational exponents are not defined when the denominator of the exponent in lowest terms is even and the base is negative.</span>
Now, let x be a negative number and n, an even number, then
But the even root of a negative number is not a real number.
for example,
Hence, <span>rational exponents are not defined when the denominator of the exponent in lowest terms is even and the base is negative.</span>
