# Waring’s analogue for fractions?

So Naturally we can combine those.

Is Every positive fraction the Sum of at most 4 positive fractions squared ? How about cubes ? Etc

$$\frac{a}{b} = \sum_{i=1}^k (\frac{x_i}{y_i})^n$$

??

Every positive integer is the sum of at most $g(k)$ $k$th powers of positive integers.
For a positive rational $\frac{m}{n}$ there is some integer $w$ such that $n | w^k.$ Indeed, for each prime $p$ dividing $n,$ with exponent $e,$ meaning $p^e | n$ but not $p^{e+1}|n,$ let the exponent $f$ of $p$ in $w$ be $f =\left\lceil \frac{e}{k} \right \rceil,$ so that $kf \geq e.$
Where was I, we have $n | w^k,$ so $$M = w^k \cdot \frac{m}{n } = m \cdot \frac{w^k}{n}$$ is an integer. Write $M$ as the sum of $g(k)$ or fewer $k$th powers, $M = x_1^k + \cdots x_r^k$ in integers, $r \leq g(k). \;$ Then $$\frac{m}{n} = \frac{M}{w^k} = \left( \frac{x_1}{w} \right)^k + \cdots + \left( \frac{x_r}{w} \right)^k$$