# [dc], 125 bytes

    15k?ddsk1-A 5^*sw1sn0[A 5^ln+_1^+ln1+dsnlw!<y]syr1<y1lk/*sz[si1[li*li1-dsi0<p]spli0<p]so0dsw[lzlw^lwlox/+lw1+dswA 2^!<b]dsbxp

Unlike the other dc answer, this works for *all* real \$x\$ greater than or equal to \$1 (\$1 ≤ x\$). Accurate to 4-5 places after the decimal.

I would have included a TIO link here, but for some reason this throws a segmentation fault with the version there (`dc 1.3`) whereas it does *not* with my local version (`dc 1.3.95`). 

### Explanation

As **dc** does not support raising numbers to non-integer exponents to calculate \$x^\frac1x\$, this takes advantage of the fact that:

$$x^\frac1x = e^\frac{\ln x}x$$

So, to calculate \$\ln(x)\$, this also takes advantage of the fact that:

$$\int \frac1x dx = \ln(x) + c$$

whose definite integral from \$1\$ to \$b = x\$ is numerically-approximated in increments of \$10^{-5}\$ using the following summation formula:

$$\int_1^b \frac1x dx = \sum_{i=1}^{10^{5}(b-1)} \frac 1 {10^5 + i}$$

The resulting sum is then multiplied by \$\frac1x\$ to get \$\frac{\ln(x)}x\$. \$e^{\frac{\ln(x)}x}\$ is then finally calculated using the \$e^x\$ Maclaurin Series to 100 terms as follows:

$$e^x=\sum^{10^2}_{n=0}\frac{x^n}{n!}$$

This results in our relatively accurate output of \$x^\frac1x\$.

[dc]: https://www.gnu.org/software/bc/manual/dc-1.05/html_mono/dc.html


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