Cube root of 175,616, no calculator

Question

175{,}616 is a perfect cube. Solve in your head. What is \sqrt[3]{175{,}616}? Aim: under 20 seconds.

Accepted Answer

Two independent reads. 1. Units digit. The cube ends in 6. In the cube units map (0	o0, 1	o1, 2	o8, 3	o7, 4	o4, 5	o5, 6	o6, 7	o3, 8	o2, 9	o9), a 6 can only come from a root ending in 6. So the last digit of the root is 6. 2. Magnitude. Strip the last three digits, leaving 175. Since 5^3 = 125 \le 175 < 216 = 6^3, the tens digit is 5. Combine: \sqrt[3]{175{,}616} = 56. Verify the anchor once: 56^3 = 56^2 	imes 56 = 3136 	imes 56 = 3136 	imes 50 + 3136 	imes 6 = 156{,}800 + 18{,}816 = 175{,}616. Takeaway. For any two-digit perfect cube: read the last digit from the units map (here…

Cube root of 175,616, no calculator

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