Cube root of 59,319, no calculator

Question

59{,}319 is a perfect cube. Solve in your head. What is \sqrt[3]{59{,}319}? Aim: under 20 seconds.

Accepted Answer

Two independent reads: 1. Units digit. Cubes end in a digit that uniquely identifies the root's last digit: \dots 9 can only come from a root ending in 9 (since 9^3 = 729). The map is: 0	o0, 1	o1, 2	o8, 3	o7, 4	o4, 5	o5, 6	o6, 7	o3, 8	o2, 9	o9, digits 2/8 and 3/7 swap, everything else is fixed. 2. Magnitude. Strip the last three digits, leaving 59. Since 3^3 = 27 \le 59 < 64 = 4^3, the tens digit is 3. Combine: \sqrt[3]{59{,}319} = 39. Verify the anchor once in your life: 39^3 = 39^2 	imes 39 = 1521 	imes 39 = 1521 	imes 40 - 1521 = 60{,}840 - 1{,}521 = 59{,}319. Takeaway. For an…

Cube root of 59,319, no calculator

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