I apologize in advance if this question is trivial to most of you but I'd like to verify if my understanding is correct. I want to verify the following rules:
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If we have $\dfrac{1}{\sqrt{x}}$, then it can be rewritten as $x\cdot x^2$ (because when the denominator $x^{1/2}$ is moved to numerator, we flip the fraction).
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$\dfrac{x}{x^2}$ can be rewritten as $x\cdot x^{-2}$ (because moving the denominator to numerator, we change the sign).
Are these mathematically correct ?
Thank you
Best Answer
Here is a rule that says you flip fractions when moving from bottom to top:
$$\frac{A/B}{\color{Blue}{C/D}} = \frac{A}{B}\times \color{Blue}{\frac{D}{C}}.$$
Notice that the fraction $C/D$ does not appear in any exponent. Your "rule"
$$ \frac{A}{B^{\color{Red}{C/D}}}=A\times B^{\color{Red}{D/C}}$$
is made-up and wrong. It is not correct. What is correct is that moving powers across the fraction bar end up changing the sign of the exponent:
$$\frac{A}{ B^C}=A\times B^{-C}.$$