I'd nuance @John Palmieri 's answer by noting that $\sqrt[n]{x}$ is a *multivalued* function of $x$, therefore ambiguous. In the (frequent) case $n=2,\ x\in\mathbb{R}^{+}$, the usual *convention* is to use the positive real root of $x$ as the sole value. This convention breaks for $ x\not\in\mathbb{R}^{+}$ since $i$ denotes *a* solution of the equation $x^2+1=0$ (which one ?)... The same ambiguity extends to cases $n>2$...

You can see the possible values of your multivalued expression that way :

```
a = 70 - 13*sqrt(29)
b = 70 + 13*sqrt(29)
t=var("t")
aa=(t^3-a).roots(ring=QQbar, multiplicities=False)# ; print("a in ", aa)
bb=(t^3-b).roots(ring=QQbar, multiplicities=False)# ; print("b in ", bb)
print ("x in ", [a+b for a, b in cartesian_product([aa, bb]).list()])
```

which prints (see there) :

```
a in [-0.1925824035672521?, 0.09629120178362601? - 0.1667812538111072?*I, 0.09629120178362601? + 0.1667812538111072?*I]
b in [5.192582403567252?, -2.596291201783626? - 4.496908272733301?*I, -2.596291201783626? + 4.496908272733301?*I]
x in [5.000000000000000?, -2.788873605350878? - 4.496908272733301?*I, -2.788873605350878? + 4.496908272733301?*I, 5.288873605350878? - 0.1667812538111072?*I, -2.500000000000000? - 4.663689526544407?*I, -2.500000000000000? + 4.330127018922193?*I, 5.288873605350878? + 0.1667812538111072?*I, -2.500000000000000? - 4.330127018922193?*I, -2.500000000000000? + 4.663689526544407?*I]
```

Sorry to muddy the waters...

HTH,

You should update your version of Sage: the syntax "print x" (rather than "print(x)") indicates that you are using a pretty old version.

You are right, I still use Sage v7.3.