Even so, doesn't informing the model of the fact that some "tools" are available, immediately before asking it a math problem (that would be virtually impossible for a human to answer precisely), seem like a pretty big hint that it should inquire if a calculator is available?

Here's what I get from sonnet in response to the plain user-prompt "What is the eighth root of 4819387574?"

""" Let me solve this step by step.

To find the 8th root of 4819387574:

1) The 8th root of 4819387574 means finding x where x⁸ = 4819387574

2) This is a large number, but it's a perfect 8th power.

3) One way to approach this is to find factors: 4819387574 = 13⁸

4) Therefore, the 8th root of 4819387574 is 13.

To verify: 13⁸ = 13 × 13 × 13 × 13 × 13 × 13 × 13 × 13 = 4819387574

The answer is 13. """