Perl 5, 72 bytes

$n=1;print substr(Math::BigFloat->bpi()**(1/$n++),0,21).$/ while($n<13);

Even using $pi$ from BigFloat, it doesn't seem to exactly match the given outputs. I'm sure it's truncated properly, though.

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Fortran (GFortran), 48 bytes

print*,(acos(-1._16)**(1/real(i,16)),i=1,12)
end

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Just an implicit loop with quad-precision vars. Not transferable, but works with GCC.

C (gcc) with libquadmath, 107

• 2 bytes saved thanks to @ceilingcat.

This uses __float128 to get the required precision.

#import<quadmath.h>
s[9];main(i){for(;i<13;puts(s))quadmath_snprintf(s,36,"%.99Qf",expq(logq(M_PIq)/i++));}

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---

I was curious to try this using the GNU MPFR library too:

C (gcc) with libmpfr, 123

• 13 bytes saved thanks to @ceilingcat.

#include<mpfr.h>
i;main(){MPFR_DECL_INIT(p,99);for(;i<12;){mpfr_const_pi(p,99);mpfr_root(p,p,++i,MPFR_RNDN);mpfr_printf("%.19Rfn",p);}}

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R + Rmpfr, 80 bytes

Requires package Rmpfr.

mpfr("1.00000000238982476721842284052415179434",30 *log2(10))^(479001600/(1:12))

Gives the following, with truncations mine and some minor formatting

12 'mpfr' numbers of precision  99   bits 
3.1415926535897932384... 
1.7724538509055160272...
1.4645918875615232630...
1.3313353638003897127...
1.2572741156691850593... 
1.2102032422537642759...
1.1776640300231973966... 
1.1538350678499894305...
1.1356352767378998683... 
1.1212823532318632987...
1.1096740829646979321...  
1.1000923789635869829...

See below for why simpler versions don't work. Direct calculation using exponents 1/N for N=1,12 returned inaccurate value fairly early. I figured that was probably due to R or Rmpfr rounding 1/3 early, so whole number exponents would be preferred. So I calculated pi^(12!) (12!=479001600) using Mathematica, then raised it to the power of 12!/N, which would always be a whole number. I had to further tune it by passing the number to Rmpfr as a character vector (so R wouldn't round it), and by using an arbitrarily high precision in both Mathematica and Rmpfr so it would truncate accurately. Because of those arbitrary additions, I can probably shave off a few bytes, but I'm good with it as is.

R, 29 bytes

This only works if R value for pi is accurate, which it isn't. Even reassigning the variable pi to a more accurate representation does not improve accuracy, as it rounds or something around 17 decimals.

format(pi^(1/1:12),nsmall=20)

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Or, for 30 bytes

options(digits=20);pi^(1/1:12)

There's a package that gives a more accurate value for pi and other floating point numbers, Rmpfr, which you'll find referenced in questions about pi in R. One might expect the following to give the desired output.

library(Rmpfr)
Const("pi",20 *log2(10))^(1/1:12)

It doesn't. It gives

12 'mpfr' numbers of precision  66   bits   
[1] 3.1415926535897932385  1.7724538509055160273  1.464591887561523232
[4] 1.3313353638003897128 1.2572741156691850754 1.2102032422537642632
[7]  1.177664030023197386 1.1538350678499894305 1.1356352767378998604
[10] 1.1212823532318633058 1.1096740829646979353 1.1000923789635869772

This is wrong on all counts by rounding or being a few off in the last digits (sidenote: the rnd.mode flag for mpfr does not fix this). Now one might think if we went up to many digits (say 100), then it would surely be correct to the first 20 digits. Nope

12 'mpfr' numbers of precision  332   bits 
 [1] 3.1415926535897932384...
 [2] 1.7724538509055160272...
 [3] 1.4645918875615232319...
 [4] 1.3313353638003897127...
 [5] 1.2572741156691850753...
 [6] 1.2102032422537642631...
 [7] 1.1776640300231973859...
 [8] 1.1538350678499894305...
 [9] 1.1356352767378998603...
[10] 1.1212823532318633058...
[11] 1.1096740829646979353...
[12] 1.1000923789635869771...

(Truncations mine). These don't all match OP or the other responses.

C (gcc -lm), 171 131 122 bytes

_{Shaved off 40 49 bytes thanks to ceilingcat.}

i;p(){for(;i<12;)printf("%.14f%d ",pow(acos(~fesetround(1024)),1./++i),L"纀罶?얡꜇???胛??"[i]);}

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Output:

3.1415926535897932384 1.7724538509055160272 1.4645918875615232630 1.3313353638003897127 1.2572741156691850593 1.2102032422537642759 1.1776640300231973966 1.1538350678499894305 1.1356352767378998683 1.1212823532318632987 1.1096740829646979321 1.1000923789635869829

Octave, 23 bytes

vpa(pi,22).^(1./(1:12))

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Declares a variable precision arithmetic (VPA) pi. Octave then cleverly infers that the double constant pi actually means pi, not whatever the double constant contains.

Java 10, 316 299 268 259 bytes

v->Math.PI+"2384 1.7724538509055160272 1.4645918875615232630 1.3313353638003897127 1.2572741156691850593 1.2102032422537642759 1.1776640300231973966 1.1538350678499894305 1.1356352767378998683 1.1212823532318632987 1.1096740829646979321 1.1000923789635869829"

-40 bytes by just hard-coding the output instead of calculating it.. >.>
-9 bytes thanks to @ceilingcat.

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Old 299 291 bytes answer with actual calculations..

v->{var P=new java.math.BigDecimal(Math.PI+"2384");var x=P;for(int i=0;++i<13;System.out.println(x)){var p=P;for(x=P.divide(P.valueOf(i),21,5);x.subtract(p).abs().compareTo(P.ONE.movePointLeft(22))>0;)x=P.valueOf(i-1).multiply(p=x).add(P.divide(x.pow(i-1),21,5)).divide(P.valueOf(i),21,5);}}

Not entirely precise, but good enough to have the first 20 digits correct.

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Explanation:

Most bytes come from the fact that BigDecimal doesn't have a builtin for BigDecimal.pow(BigDecimal) nor the $n$th root, so we'll have to calculate this manually..

v->{                         // Method with empty unused parameter and no return-type
  var P=new java.math.BigDecimal(Math.PI+"2384");
                             //  Create a BigDecimal for PI
  var x=P;                   //  Create a BigDecimal to print after every iteration
  for(int i=0;++i<13;        //  Loop `i` in the range [1,12]:
      System.out.println(x)){//    After every iteration: print `x` to STDOUT
    var p=P;                 //   Create a BigDecimal to save the previous value
    for(x=P.divide(P.valueOf(i),
                             //   Set `x` to PI divided by `i`
                   21,5);    //   (with precision 21 and rounding mode HALF_DOWN)
                             //   Continue an inner loop as long as:
        x.subtract(p).abs()  //   The absolute difference between `x` and `p`
         .compareTo(P.ONE.movePointLeft(22))>0;)
                             //   is larger than 1e-22
      x=                     //    Set `x` to:
        P.valueOf(i-1)       //     `i-1`
         .multiply(p=x)      //     Multiplied by `x` (and store the previous `x` in `p`)
         .add(               //     And add:
          P.divide(          //      PI divided by
           x.pow(i-1),21,5)) //      `x` to the power `i-1`
         .divide(P.valueOf(i),21,5);}}
                             //     Divided by `i`

Python 3, 61 69 63 bytes

Variant of Jitse's answer, who wants to stick to standard libraries. As mentioned by @Seb & @Jitse, Rational or E/E are needed because 1/i isn't precise enough as float.

from sympy import*;i=E/E
while i<13:print(N(pi**(1/i),99));i+=1

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As a bonus, sympy allows to output 99 decimals with the same byte count as for 20 decimals:

3.14159265358979323846264338327950288419716939937510582097494459230781640628620899862803482534211707
1.77245385090551602729816748334114518279754945612238712821380778985291128459103218137495065673854467
1.46459188756152326302014252726379039173859685562793717435725593713839364979828626614568206782035382
1.33133536380038971279753491795028085330936622381810425845370748286670076101723561496824589105670695
1.25727411566918505938452211411044829390616631003965817353418162262716270074393519788994784267497245
1.21020324225376427596603076175994353105276255513631810467305643780646240044814351479113420061960303
1.17766403002319739668470085583704641096763145003350008756991802723553566109741289228117335907046316
1.15383506784998943054096521314988190017738887987082462087415032873413337243208073328092932090341440
1.13563527673789986837981146453309184806238366027985302804182074656192075351096762716281170420998022
1.12128235323186329872203095522015520934269381296656043824699758762377264153679046528721553289724806
1.10967408296469793211291254568401340513968966612239757917494488030006513039871082996506101420959919
1.10009237896358698298222864697784031503277459309498017808547779493972810083028975298933600450861044

Ruby -rbigdecimal/math, 65 50 bytes

Conveniently, even though BigMath.PI(9) only guarantees precision up to 9 digits, it actually is precise up to 26, which is enough to calculate the exponents, which use the builtin Rational fractions instead of floats to ensure the precision is still good enough. (Also, making a Rational saves a byte over dividing it normally, since Ruby uses integer division if both arguments are integers, necessitating the use of 1.0 somewhere in the code.)

-15 bytes from histocrat!

1.upto(12){|i|puts BigMath.PI(9).**(1r/i).to_s ?F}

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Jelly, 46 bytes

12İ€ØP*×ȷ19Ḟ+“Ṿẏ⁽)¬ọƓỴ³ɲỊUị&ıİḣl’ḃ4ȷ¤_2ȷD;"€”.

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Similar to some other answers, but encodes the difference between the Jelly answer and the correct answer (with big integer arithmetic). Full explanation to follow.

Python 3, 146 bytes

import math
for i in range(1,13):print(str(math.pi**(1/i))[:~(i==2)]+str([32384,60272,2630,7127,'0593',2759,3966,4305,8683,2987,9321,69829][i-1]))

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This uses a similar method to Arnauld's JavaScript solution, by using the inbuilt pi value with the extra precision added to the end.

cQuents, 268 bytes

#1&"3.1415926535897932384
1.7724538509055160272
1.4645918875615232630
1.3313353638003897127
1.2572741156691850593
1.2102032422537642759
1.1776640300231973966
1.1538350678499894305
1.1356352767378998683
1.1212823532318632987
1.1096740829646979321
1.1000923789635869829"

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There is no way to get more precision out of floats in cQuents, so the values must be hardcoded as strings.

cQuents, 11 bytes

#12&`p^(1/$

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Does not reach the required precision levels.

bc -l, 28

for(;i<12;)e(l(4*a(1))/++i)

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Canvas, 30 bytes

６«｛“≥αｙＨT.─Ｃ¹„1.0000ŗ┤“^m„┘÷＾］

Don't try it here! It'll take a while to calculate all those digits (it took ~15 minutes to run for me). Rather, here's a version of the same code, only outputting the last 3 items.

Computes C ^ (27720/N) where C is the hard-coded constant pi^(1/27720) = 1.000041297024626834690309 and N is looped over from 1 to 12. The big number library decided to expand the amount of significant digits for successively bigger N, making the code take unreasonable amounts of time to run.

Julia 1.0, 18 bytes

@.π^(1/big(1:12))

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(1./big(1:12)) divides a BigInt 1 by each of 1 thru 12, then π.^ raises pi to each of those values. So long as there is one BigInt or BigFloat involved in each computation, it will calculate the result at that precision. The @. macro transforms the code to add dots to every function call (thus the dots that appear in my explanation that don't appear in the code snippet), this causes it to "broadcast" which for this purpose means do it all elementwise.

20->19 thanks to Robin Ryder

19->18 thanks to TimD

Bash + bc + coreutils, 51, 40, 39, 36 bytes

following @ChristianSievers comment |cut -c -21 could be removed.

-3 bytes thanks to @DigitalTrauma.

echo "e(l(4*a(1))/"{1..12}");"|bc -l

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Some explanations

• bc -l define math functions and set scale to 20, see man bc for more details
• a() atan function, so 4*a(1) is pi
• e() exp function
• l() log function, so e(l(x)/y) is x^(1/y)

Wolfram Language (Mathematica), 22 20 19 bytes

-2 bytes thanks to game0ver -1 byte thanks to LegionMammal978

Pi^(1`11/Range@12)&

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Pari/GP, 20 bytes

vector(12,n,Pi^n^-1)

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(TIO needs some supporting code, but this works as is in the REPL.

05AB1E (legacy), 47 43 40 bytes

•u¬Œo¡õ≠ÎĆζw1Á4¼©éßw–xùó1O•5ôεžqN>zm16£ì

-7 bytes thanks to @Grimy.

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Uses the legacy version, because for some reason the new version outputs $1.0$ for $pi^{frac{1}{1}}$.. :S

Explanation:

•u¬Œo¡õ≠ÎĆζw1Á4¼©éßw–xùó1O•
             # Push compressed integer 323846027232630971275059342759739669430598683329877932169829
 5ô          # Split it into parts of size 5:
             #  [32384,60272,32630,97127,50593,42759,73966,94305,98683,32987,79321,69829]
ε            # Then map each integer to:
   N>        #  Take the 0-based map-index, and increase it by 1
     z       #  Calculate 1/(index+1)
 žq   m      #  Then calculate PI the power of this
       16£   #  Only leave the first 16 characters (including decimal dot)
          ì  #  And prepend it before the current integer we're mapping
             # (after which the mapped result is output implicitly)

See this 05AB1E tip of mine (sections How to compress large integers?) to understand why •u¬Œo¡õ≠ÎĆζw1Á4¼©éßw–xùó1O• is 323846027232630971275059342759739669430598683329877932169829.

Python 3, 91 bytes

import fractions as f;p=f.Decimal('%s2384'%f.math.pi);i=p/p
while i<13:print(p**i**-1);i+=1

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-23 bytes thanks to flornquake

JavaScript (ES7), 121 bytes

Computes as many digits as the precision of IEEE 754 allows and hardcodes the other ones.

_=>[32384,60272,2630,7127,[n=0]+593,2759,3966,4305,8683,2987,9321,69829].map(v=>(Math.PI**(1/++n)+'').slice(0,~(n==2))+v)

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Commented

_ =>                          // input is ignored
[ 32384, 60272, 2630, 7127,   // hard-coded digits for n=1 to n=4
  [n = 0] + 593,              // initialize n to 0, and set this entry to '0593' (n=5)
  2759, 3966, 4305, 8683,     // hard-coded digits for n=6 to n=9
  2987, 9321, 69829           // hard-coded digits for n=10 to n=12
].map(v =>                    // for each entry in the above array:
  (Math.PI ** (1 / ++n) + '') //   increment n; compute π**(1/n) and coerce it to a string
  .slice(0, ~(n == 2))        //   remove the last 2 digits if n=2,
                              //   or only the last digit otherwise
  + v                         //   append the hard-coded digits
)                             // end of map()

APL (Dyalog Extended), 9 8 bytes^SBCS

(⍳12)√○1

Try it online! (⎕PP←34 is Print Precision: 34 digits; ⎕FR←1287 is Float Representation: 128-bit decimal)

○1 $π×1$

(…)√ take the following roots of that:

 ⍳12 ɩndices 1 through 12