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The trick, step by step
Look at the last digit: 5476 ends in 6, so the root ends in 4 or 6 (4² = 16, 6² = 36).
Drop the last two digits: 54. 7² = 49 is the biggest square not above 54, so the root starts with 7.
Pick the ending: 75² = 5625 is bigger than 5476, so take the smaller one: 74.
given working answer taken away
Why it works
A square's last digit depends only on the root's last digit, and each ending (except 0 and 5) comes from two digits that add to 10. The digits left after dropping two tell the tens digit, and comparing with the square of the number ending in 5 picks between the two endings.
Practise this trick