Math Motives

In the post viewtopic.php?t=8 we talked about the Laws for powers:

Laws of Powers: let \(a > 0, b > 0\) be two positive numbers, \(m, n\) be two integers. Then

- Products of Powers: \(a^{m} \cdot a^{n} = a^{m + n}\);
- Powers of Powers: \( (a^{m})^{n} = a^{m \cdot n}\);
- Powers of Products: \( (a \cdot b)^{n} = a^n \cdot b^n \).

Note that these laws can be extended to

- negative bases (with integral powers), like \((-1)^2 = 1, (-1)^3 = -1\);
- fractional powers (of positive baese), like \( 4^{1/2} = 2, 8^{2/3} = 4\).

What happens if we allow fractional powers of negative bases? Consider the example with \(a=b=-1, m=n=\frac{1}{2}\).

We want to simplify \((-1)^{1/2}\cdot (-1)^{1/2}\).

- Attempt: apply the Law for Products of Powers: \( (-1)^{1/2}\cdot (-1)^{1/2} = (-1)^{1/2 + 1/2} = (-1)^1 = -1\). Nice.
- Check: apply the Law for Powers of Powers: \( (-1)^{1/2}\cdot (-1)^{1/2} = ((-1)^{1/2})^2 =(-1)^{1/2\cdot 2} = (-1)^1 = -1\). Confirmed.
- Double check: apply the Law for owers of Products: \( (-1)^{1/2}\cdot (-1)^{1/2} = ((-1)\cdot (-1))^{1/2}=1^{1/2} =1 \). Wait.

Did we just prove \( -1 = (-1)^{1/2}\cdot (-1)^{1/2} = 1 \)? What went wrong?