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If you divide an number by another numbers reciprocal

i.e

x / (1 / y)

Can this be expressed in another way

is dividing by the reciprocal equivelant to another operation?

3 Answers 3

10

Yes. $\frac{x}{\frac{1}{y}} = xy$.

This is a simple consequence of two things: $x = \frac{x}{1}$, and $\frac{\quad\frac{a}{b}\quad}{\frac{x}{y}} = \frac {ay}{bx}.$ The latter is just the usual rule for dividing fractions, written in fraction form.

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The only possible issue with writing $\frac{x}{\frac{1}{y}}$ as $xy$ is that you lose the information that $y\ne0$. So if you are thinking about them as functions then they aren't quite the same. But whenever $y\ne 0$, $\frac{x}{\frac{1}{y}}=xy$ as noted in the other answers.

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HINT $\ $ "Integralize" the denominator, i.e. multiply by $\rm\ y\ $ the denominator (and numerator).
This is analogous to "rationalizing" a denominator $\rm\:b\:$ by multiplying by its conjugate \rm\:b'\:, i.e. \rm\displaystyle \frac{a}{b} = \frac{a\:b'}n,\ \ n = b\:b'\in\mathbb Q\:.