diff --git a/src/sage/rings/laurent_series_ring_element.pyx b/src/sage/rings/laurent_series_ring_element.pyx
index 6958a0e45a0..79eb5adeba4 100644
--- a/src/sage/rings/laurent_series_ring_element.pyx
+++ b/src/sage/rings/laurent_series_ring_element.pyx
@@ -1359,7 +1359,9 @@ cdef class LaurentSeries(AlgebraElement):
 
         - ``n`` -- integer
 
-        - ``prec`` -- integer (optional) - precision of the result
+        - ``prec`` -- integer (optional) - precision of the result. Though, if
+          this series has finite precision, then the result can not have larger
+          precision.
 
         EXAMPLES::
 
diff --git a/src/sage/rings/polynomial/polynomial_element.pyx b/src/sage/rings/polynomial/polynomial_element.pyx
index 8d67be4b531..c78f6fddb6d 100644
--- a/src/sage/rings/polynomial/polynomial_element.pyx
+++ b/src/sage/rings/polynomial/polynomial_element.pyx
@@ -9673,6 +9673,16 @@ cdef class Polynomial(CommutativeAlgebraElement):
         lowest degree is not invertible in the base ring. In both cases an
         ``ArithmeticError`` is raised.
 
+        INPUT:
+
+        - ``n`` -- positive integer; the exponent of the root
+
+        - ``prec`` -- positive integer; the precision of the result
+
+        - ``start`` -- optional; the first term of the result. This
+          is only considered when the valuation is zero, i.e. when the
+          polynomial has a nonzero constant term.
+
         .. ALGORITHM::
 
             Let us denote by `a` the polynomial from which we wish to extract
diff --git a/src/sage/rings/power_series_ring_element.pyx b/src/sage/rings/power_series_ring_element.pyx
index 1903dc4f259..4195ab0f5ed 100644
--- a/src/sage/rings/power_series_ring_element.pyx
+++ b/src/sage/rings/power_series_ring_element.pyx
@@ -1415,7 +1415,9 @@ cdef class PowerSeries(AlgebraElement):
 
         - ``n`` -- integer
 
-        - ``prec`` -- integer (optional) - the precision of the result
+        - ``prec`` -- integer (optional) - precision of the result. Though, if
+          this series has finite precision, then the result can not have larger
+          precision.
 
         EXAMPLES::
 
@@ -1426,7 +1428,7 @@ cdef class PowerSeries(AlgebraElement):
             sage: (1 + x + O(x^5)).nth_root(5)
             1 + 1/5*x - 2/25*x^2 + 6/125*x^3 - 21/625*x^4 + O(x^5)
 
-        Check that the result are consistent with taking log and exponential::
+        Check that the results are consistent with taking log and exponential::
 
             sage: R.<x> = PowerSeriesRing(QQ, default_prec=100)
             sage: p = (1 + 2*x - x^4)**200