Cube root of 32,768, no calculator

Question

32{,}768 is a perfect cube. Solve in your head. What is \sqrt[3]{32{,}768}? Aim: under 20 seconds.

Accepted Answer

Two independent reads. 1. Units digit. The cube ends in 8. In the cube units map, 8 comes only from a root ending in 2 (since 2^3 = 8). Recall the digits 2 and 8 swap under cubing (2^3 ends in 8, 8^3 = 512 ends in 2). So the root ends in 2. 2. Magnitude. Strip the last three digits, leaving 32. Since 3^3 = 27 \le 32 < 64 = 4^3, the tens digit is 3. Combine: \sqrt[3]{32{,}768} = 32. Verify the anchor once: 32^3 = 32^2 	imes 32 = 1024 	imes 32 = 32{,}768 (and indeed 32 = 2^5, so 32^3 = 2^{15} = 32{,}768). Takeaway. The 2\leftrightarrow 8 and 3\leftrightarrow 7 swaps are the only non-fixed poi…

Cube root of 32,768, no calculator

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