What is 3 to the 10th power?

Question

No calculator. Solve in your head. What is 3^{10}? Aim: under 20 seconds.

Accepted Answer

Split the exponent and square: 3^{10} = (3^{5})^2 = 243^2. Square 243: (240 + 3)^2 = 240^2 + 2 \cdot 240 \cdot 3 + 9 = 57{,}600 + 1{,}440 + 9 = 59{,}049. So 3^{10} = 59{,}049 \approx 59{,}000. Takeaway. 3^{5} = 243 is worth knowing outright; then 3^{10} is a single squaring. The clean landmark to carry is 3^{10} \approx 59{,}000 (just under 6 	imes 10^4), which anchors quick estimates of other powers of three.

What is 3 to the 10th power?

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