How many digits in 2^30

Question

pierre · Accepted Answer

2 ^ 10 = 1024 therefore 1000 < 2 ^ 10 < 2000
                         therefore 1000 ^ 3 < 2^30 < 2000 ^3 
                         therefore  10 ^9 < 2 ^30 < 8 * 10 ^9 
                        10 ^ 9 and  8 * 10 ^9 have 10 digits each 
                            therefore so has 2 ^30

Michael · Answer

It helps/is absolutely necessary to know that 2^10~1,000 (1024 to be exact, but this is unnecessary).  Thus 2^30==(2^10)^3==(1,000)^3=~1,000,000,000 (1,073,741,824 to be exact, but completely unnecessary), so there are 10 digits.

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