$b^{\frac{m}{n}}=(b^{\frac{1}{n}})^m=(b^m)^{\frac{1}{n}}$ except $b$ is not negative when $n$ is Even.

Question

The following property, known as Rational number property, is taken from the book (I am following now a days) College Algebra by Raymond A Barnett and Micheal R Ziegler

I restate,

$b^{\frac{m}{n}}=(b^{\frac{1}{n}})^m=(b^m)^{\frac{1}{n}}$ except $b$ is not negative when $n$ is Even.

Note: The restrictions are given only for $b$ and $n$ and not for $m$.

I tried working against the restrictions (given in book) but since there is no restriction given for $m$, I took value of $m$ arbitrarily. Even though things are going right.

$$((-1)^3)^{\frac{1}{2}} = ((-1)^{\frac{1}{2}})^3$$

$$\sqrt{(-1)^3}=(\sqrt{-1})^3$$

$$\sqrt{-1}=\sqrt{-1} \sqrt{-1} \sqrt{-1}$$

Note: I know that $\sqrt{a}\sqrt{b}=\sqrt{ab}$ only and only if, $a \geq 0$ or $b \geq 0$ but here I ignored this restriction since this time I am only verifying the restrictions given in my book. There is no such restriction given in my book $\sqrt{a}\sqrt{b}=\sqrt{ab}$ only and only if, $a \geq 0$ or $b \geq 0$

So ignoring the above restriction we get,

$$\sqrt{-1}=\sqrt{-1 \times -1} \sqrt{-1}$$

And ultimately we get,

$$i=i$$

Hence the restrictions given in my book are disproved. The questions are,

1-I am thinking that there should has been restriction on $m$ along with $b$ and $n$.

2-Or, following restriction should has been given,

$\sqrt{a}\sqrt{b}=\sqrt{ab}$ only and only if, $a \geq 0$ or $b \geq 0$

[excuse for English]

Am i right?

Answer 1

The quoted sentence says "if the conditions are fulfilled, then the claim holds." That's known as a sufficient condition. It does not say "if the conditions are not fulfilled, then the claim doesn't hold". That would be a necessary condition. Since only the sufficient condition was claimed, the only way to disprove the claim would be to find a case where the conditions are fulfilled but the property doesn't hold.

What you did disprove by counterexample was the (not claimed) assumption that the condition is necessary. That is, there are cases which do not fulfil the condition but still the property holds.

However there are simpler counterexamples for the claim being necessary; for example, $m=n=1$ is an obvious counterexample (violating the condition that $n$ is even, but clearly $b^{\frac{1}{1}}=(b^{\frac11})^1=(b^1)^{\frac11}$ for any $b$).