diff --git a/src/core/include/mp-units/framework/expression_template.h b/src/core/include/mp-units/framework/expression_template.h
index c0ec574b1..72a2db0e2 100644
--- a/src/core/include/mp-units/framework/expression_template.h
+++ b/src/core/include/mp-units/framework/expression_template.h
@@ -334,7 +334,7 @@ struct expr_fractions : decltype(expr_fractions_impl<OneType, type_list<Ts...>>(
 
 // expr_make_spec
 template<typename NumList, typename DenList, typename OneType, template<typename...> typename To>
-[[nodiscard]] consteval auto expr_make_spec_impl()
+[[nodiscard]] MP_UNITS_CONSTEVAL auto expr_make_spec_impl()
 {
   constexpr std::size_t num = type_list_size<NumList>;
   constexpr std::size_t den = type_list_size<DenList>;
@@ -359,7 +359,7 @@ template<typename NumList, typename DenList, typename OneType, template<typename
  */
 template<typename NumList, typename DenList, typename OneType, template<typename, typename> typename Pred,
          template<typename...> typename To>
-[[nodiscard]] consteval auto get_optimized_expression()
+[[nodiscard]] MP_UNITS_CONSTEVAL auto get_optimized_expression()
 {
   using num_list = expr_consolidate<NumList>;
   using den_list = expr_consolidate<DenList>;
@@ -380,7 +380,7 @@ template<typename NumList, typename DenList, typename OneType, template<typename
  */
 template<template<typename...> typename To, typename OneType, template<typename, typename> typename Pred, typename Lhs,
          typename Rhs>
-[[nodiscard]] consteval auto expr_multiply(Lhs, Rhs)
+[[nodiscard]] MP_UNITS_CONSTEVAL auto expr_multiply(Lhs, Rhs)
 {
   if constexpr (is_same_v<Lhs, OneType>) {
     return Rhs{};
diff --git a/src/core/include/mp-units/framework/magnitude.h b/src/core/include/mp-units/framework/magnitude.h
index 8fb22eb81..d16325283 100644
--- a/src/core/include/mp-units/framework/magnitude.h
+++ b/src/core/include/mp-units/framework/magnitude.h
@@ -307,6 +307,119 @@ template<typename T>
   return checked_square(int_power(base, exp / 2));
 }
 
+template<typename T>
+[[nodiscard]] consteval std::optional<T> checked_int_pow(T base, std::uintmax_t exp)
+{
+  T result = T{1};
+  while (exp > 0u) {
+    if (exp % 2u == 1u) {
+      if (base > std::numeric_limits<T>::max() / result) {
+        return std::nullopt;
+      }
+      result *= base;
+    }
+
+    exp /= 2u;
+
+    if (base > std::numeric_limits<T>::max() / base) {
+      return (exp == 0u) ? std::make_optional(result) : std::nullopt;
+    }
+    base *= base;
+  }
+  return result;
+}
+
+template<typename T>
+[[nodiscard]] consteval std::optional<T> root(T x, std::uintmax_t n)
+{
+  // The "zeroth root" would be mathematically undefined.
+  if (n == 0) {
+    return std::nullopt;
+  }
+
+  // The "first root" is trivial.
+  if (n == 1) {
+    return x;
+  }
+
+  // We only support nontrivial roots of floating point types.
+  if (!std::is_floating_point<T>::value) {
+    return std::nullopt;
+  }
+
+  // Handle negative numbers: only odd roots are allowed.
+  if (x < 0) {
+    if (n % 2 == 0) {
+      return std::nullopt;
+    } else {
+      const auto negative_result = root(-x, n);
+      if (!negative_result.has_value()) {
+        return std::nullopt;
+      }
+      return static_cast<T>(-negative_result.value());
+    }
+  }
+
+  // Handle special cases of zero and one.
+  if (x == 0 || x == 1) {
+    return x;
+  }
+
+  // Handle numbers bewtween 0 and 1.
+  if (x < 1) {
+    const auto inverse_result = root(T{1} / x, n);
+    if (!inverse_result.has_value()) {
+      return std::nullopt;
+    }
+    return static_cast<T>(T{1} / inverse_result.value());
+  }
+
+  //
+  // At this point, error conditions are finished, and we can proceed with the "core" algorithm.
+  //
+
+  // Always use `long double` for intermediate computations.  We don't ever expect people to be
+  // calling this at runtime, so we want maximum accuracy.
+  long double lo = 1.0;
+  long double hi = static_cast<long double>(x);
+
+  // Do a binary search to find the closest value such that `checked_int_pow` recovers the input.
+  //
+  // Because we know `n > 1`, and `x > 1`, and x^n is monotonically increasing, we know that
+  // `checked_int_pow(lo, n) < x < checked_int_pow(hi, n)`.  We will preserve this as an
+  // invariant.
+  while (lo < hi) {
+    long double mid = lo + (hi - lo) / 2;
+
+    auto result = checked_int_pow(mid, n);
+
+    if (!result.has_value()) {
+      return std::nullopt;
+    }
+
+    // Early return if we get lucky with an exact answer.
+    if (result.value() == x) {
+      return static_cast<T>(mid);
+    }
+
+    // Check for stagnation.
+    if (mid == lo || mid == hi) {
+      break;
+    }
+
+    // Preserve the invariant that `checked_int_pow(lo, n) < x < checked_int_pow(hi, n)`.
+    if (result.value() < x) {
+      lo = mid;
+    } else {
+      hi = mid;
+    }
+  }
+
+  // Pick whichever one gets closer to the target.
+  const auto lo_diff = x - checked_int_pow(lo, n).value();
+  const auto hi_diff = checked_int_pow(hi, n).value() - x;
+  return static_cast<T>(lo_diff < hi_diff ? lo : hi);
+}
 
 template<typename T>
 [[nodiscard]] consteval widen_t<T> compute_base_power(MagnitudeSpec auto el)
@@ -317,9 +430,6 @@ template<typename T>
   // Note that since this function should only be called at compile time, the point of these
   // terminations is to act as "static_assert substitutes", not to actually terminate at runtime.
   const auto exp = get_exponent(el);
-  if (exp.den != 1) {
-    std::abort();  // Rational powers not yet supported
-  }
 
   if (exp.num < 0) {
     if constexpr (std::is_integral_v<T>) {
@@ -329,8 +439,19 @@ template<typename T>
     }
   }
 
-  auto power = exp.num;
-  return int_power(static_cast<widen_t<T>>(get_base_value(el)), power);
+  const auto pow_result =
+    checked_int_pow(static_cast<widen_t<T>>(get_base_value(el)), static_cast<std::uintmax_t>(exp.num));
+  if (pow_result.has_value()) {
+    const auto final_result =
+      (exp.den > 1) ? root(pow_result.value(), static_cast<std::uintmax_t>(exp.den)) : pow_result;
+    if (final_result.has_value()) {
+      return final_result.value();
+    } else {
+      std::abort();  // Root computation failed.
+    }
+  } else {
+    std::abort();  // Power computation failed.
+  }
 }
 
 // A converter for the value member variable of magnitude (below).
diff --git a/test/runtime/CMakeLists.txt b/test/runtime/CMakeLists.txt
index 0be597b38..d79ed54b3 100644
--- a/test/runtime/CMakeLists.txt
+++ b/test/runtime/CMakeLists.txt
@@ -23,8 +23,14 @@
 find_package(Catch2 3 REQUIRED)
 
 add_executable(
-    unit_tests_runtime distribution_test.cpp fixed_string_test.cpp fmt_test.cpp math_test.cpp atomic_test.cpp
-                       truncation_test.cpp
+    unit_tests_runtime
+    distribution_test.cpp
+    fixed_string_test.cpp
+    fmt_test.cpp
+    math_test.cpp
+    atomic_test.cpp
+    truncation_test.cpp
+    quantity_test.cpp
 )
 if(${projectPrefix}BUILD_CXX_MODULES)
     target_compile_definitions(unit_tests_runtime PUBLIC ${projectPrefix}MODULES)
diff --git a/test/runtime/quantity_test.cpp b/test/runtime/quantity_test.cpp
new file mode 100644
index 000000000..151845a7a
--- /dev/null
+++ b/test/runtime/quantity_test.cpp
@@ -0,0 +1,75 @@
+// The MIT License (MIT)
+//
+// Copyright (c) 2024 Chip Hogg
+//
+// Permission is hereby granted, free of charge, to any person obtaining a copy
+// of this software and associated documentation files (the "Software"), to deal
+// in the Software without restriction, including without limitation the rights
+// to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
+// copies of the Software, and to permit persons to whom the Software is
+// furnished to do so, subject to the following conditions:
+//
+// The above copyright notice and this permission notice shall be included in all
+// copies or substantial portions of the Software.
+//
+// THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
+// IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
+// FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
+// AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
+// LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
+// OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
+// SOFTWARE.
+
+#include <catch2/catch_test_macros.hpp>
+#ifdef MP_UNITS_IMPORT_STD
+import std;
+#else
+#include <atomic>
+#include <numbers>
+#endif
+#ifdef MP_UNITS_MODULES
+import mp_units;
+#else
+#include <mp-units/math.h>
+#include <mp-units/systems/si.h>
+#endif
+
+using namespace mp_units;
+using namespace mp_units::si::unit_symbols;
+
+namespace {
+
+template<typename T>
+constexpr bool within_4_ulps(T a, T b)
+{
+  static_assert(std::is_floating_point_v<T>);
+  auto walk_ulps = [](T x, int n) {
+    while (n > 0) {
+      x = std::nextafter(x, std::numeric_limits<T>::infinity());
+      --n;
+    }
+    while (n < 0) {
+      x = std::nextafter(x, -std::numeric_limits<T>::infinity());
+      ++n;
+    }
+    return x;
+  };
+
+  return (walk_ulps(a, -4) <= b) && (b <= walk_ulps(a, 4));
+}
+
+}  // namespace
+
+// conversion requiring radical magnitudes
+TEST_CASE("unit conversions support radical magnitudes", "[conversion][radical]")
+{
+  REQUIRE(within_4_ulps(sqrt((1.0 * m) * (1.0 * km)).numerical_value_in(m), sqrt(1000.0)));
+}
+
+// Reproducing issue #474 exactly:
+TEST_CASE("Issue 474 is fixed", "[conversion][radical]")
+{
+  constexpr auto val_issue_474 = 8.0 * si::si2019::boltzmann_constant * 1000.0 * K / (std::numbers::pi * 10 * Da);
+  REQUIRE(within_4_ulps(sqrt(val_issue_474).numerical_value_in(m / s),
+                        sqrt(val_issue_474.numerical_value_in(m * m / s / s))));
+}
diff --git a/test/static/quantity_test.cpp b/test/static/quantity_test.cpp
index 9b3abbb0a..d8ddfe609 100644
--- a/test/static/quantity_test.cpp
+++ b/test/static/quantity_test.cpp
@@ -199,7 +199,6 @@ static_assert(std::convertible_to<quantity<isq::length[km], int>, quantity<isq::
 static_assert(std::constructible_from<quantity<isq::length[km]>, quantity<isq::length[m], int>>);
 static_assert(std::convertible_to<quantity<isq::length[m], int>, quantity<isq::length[km]>>);
 
-
 ///////////////////////
 // obtaining a number
 ///////////////////////