Cube root of 300,763, no calculator

Question

300{,}763 is a perfect cube. Solve in your head. What is \sqrt[3]{300{,}763}? Aim: under 20 seconds.

Accepted Answer

Two independent reads. 1. Units digit. The cube ends in 3. In the cube units map, 3 comes only from a root ending in 7 (since 7^3 = 343 ends in 3). This is the 3\leftrightarrow 7 swap. So the root ends in 7. 2. Magnitude. Strip the last three digits, leaving 300. Since 6^3 = 216 \le 300 < 343 = 7^3, the tens digit is 6. Combine: \sqrt[3]{300{,}763} = 67. Verify the anchor once: 67^3 = 67^2 	imes 67 = 4489 	imes 67 = 4489 	imes 70 - 4489 	imes 3 = 314{,}230 - 13{,}467 = 300{,}763. Takeaway. "Ends in 3" instantly gives "root ends in 7" via the 3\leftrightarrow 7 swap, and bracketing the lea…

Cube root of 300,763, no calculator

Your answer

More Mental Math questions