What is 1.5 to the 8th power?

Question

No calculator. Solve in your head. Estimate 1.5^{8} to the nearest whole number. Aim: under 30 seconds.

Accepted Answer

An exponent of 8 is three squarings (8 = 2^3): 1.5^2 = 2.25, \qquad 2.25^2 = 5.0625, \qquad 5.0625^2 \approx 25.63. For the last square, 5.0625^2 \approx 5.06^2 = 25.6. So 1.5^{8} \approx 25.6 (exact: 25.629). Takeaway. When the exponent is a power of two, reach the answer by squaring repeatedly rather than multiplying eight times, three multiplications instead of seven. The same trick handles x^{4} (two squarings) and x^{16} (four).

What is 1.5 to the 8th power?

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