Cube root of 704,969, no calculator

Question

704{,}969 is a perfect cube. Solve in your head. What is \sqrt[3]{704{,}969}? Aim: under 20 seconds.

Accepted Answer

Two independent reads. 1. Units digit. The cube ends in 9. In the cube units map, 9 comes only from a root ending in 9 (since 9^3 = 729 ends in 9). It is a fixed point. So the root ends in 9. 2. Magnitude. Strip the last three digits, leaving 704. Since 8^3 = 512 \le 704 < 729 = 9^3, the tens digit is 8. Combine: \sqrt[3]{704{,}969} = 89. Verify the anchor once: 89^3 = 89^2 	imes 89 = 7921 	imes 89 = 7921 	imes 90 - 7921 = 712{,}890 - 7{,}921 = 704{,}969. Takeaway. For a two-digit cube, the units digit read (9 	o 9) and the magnitude read (bracket 704 between 8^3 and 9^3) together pin the…

Cube root of 704,969, no calculator

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