Estimate 10^0.1 without using a calculator. How many digits can you get? Show me your working.
by 
Quirkie
ANSWERS: 9
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Hmm.. this is 1dB. 3dB is 2, so this is maybe around 1.2?
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Off the top of my head I would ballpark 1.025. There was no work, it was just using my head.
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Repeatly taking square roots, the 4th root of 10 is about 1.8, the 8th root about 1.4, the 16 root is about 1.2. So the tenth root would be about 1.3
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You could also do it backwards, I suppose. x=10^0.1 Which is also means x^10 = 10 x^2 ) ^5 = 10 and then graph it? Or use an iteriation process maybe? I don't know, i've just woken up and its too early to actually do anything other than give theory
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yes i definitely came on this site to do some math problems...what the hell is going on with you people?
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Here's my attempt: 2^10 is 1024 which is roughly 1000 = 10^3 10^3 ~= 2^10 10^0.3 ~= 2^1 10^2.1 ~= 2^7 10^0.1 ~= 2^7/10^2 = 1.28 since 10^3 is less than 2^10, 10^0.1 must be less than 1.28 But 10^0.1 must be greater than 1.2: 3^10 = 9^5 = 9*9^4 < 9*10^4 6^10 < 2^10 * 9 * 10^4 12^10 < 2^20 * 9 * 10^4 1.2^10 < 2^20 * 9 / 10^6 = 9*(1.024)^2 < 10 1.2 < 10^0.1 So that gets it between 1.2 and 1.28 I have a feeling you can do better though.
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It has been shown that 10^x is between 1.2 and 1.28. Given that 1.25 falls within this range and is 5^3/10^2 we can manipulate this to narrow the range of answers. Copying Quirkie's method a little: 5^10 = (5^5)^2 = ((5^2)(5^3))^2 = (25 * 125)^2 = 3125^2 = 9765625 < 10^7 (but not much less) (I did this on paper using long multiplication so not cheating!) Then 5 < 10^0.7 5^3 < 10^2.1 1.25 < 10^0.1 and we can say that 10^0.1 lies between 1.25 and 1.28.
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This also means that unknown^10 = 10 1^10 = 1 and 2^10 = 1024, thus answer is between 1 and 2, and probably closer to 1 than to 2 because 10 is closer to 1 than 1024. So we know that answer equals 1 plus a fraction (which is probably less than ½). 1.1^2 = 1.21 1.1^4 = (1.1^2)^2 = 1.21^2 = 1.4641 which is ï‚» sqrt(2) so 1.1^8 = (1.1^4)^2 ï‚» (sqrt(2))^2 ï‚» 2 and 1.1^10 = 1.1^8 × 1.1^2 ï‚» 2 × 1.21 ï‚» 2½ so 1.1 is too low. 1.2^2 = 1.44 which is ï‚» sqrt(2) so 1.2^4 = (1.2^2)^2 ï‚» (sqrt(2))^2 ï‚» 2 and 1.2^8 = (1.2^4)^2 ï‚» 2^2 = 4 1.2^10 = 1.2^8 × 1.2^2 ï‚» 4 × 1.44 = 5.76 so 1.2 is also too low. 1.3^2 = 1.69 ï‚» 1.7 1.3^4 = (1.3^2)^2 ï‚» 1.7^2 = 2.89 ï‚» 2.9 1.3^8 = (1.3^4)^2 ï‚» 2.9^2 = 8.41 ï‚» 8.4 1.3^10 = 1.3^8 × 1.3^2 ï‚» 8.4 × 1.7 ï‚» 8 × 2 ï‚» 16 so 1.3 is too high. 1.25 = 5/4 1.25^2 = (5/4)^2 = 25/16 1.25^4 = (1.25^2)^2 = (25/16)^2 = 625/256 ï‚» 2.5 = 5/2 1.25^8 = (1.25^4)^2 ï‚» 2.5^2 = (5/2)^2 = 25/4 = 6.25 1.25^10 = 1.25^8 × 1.25^2 ï‚» (25/4) × (25/16) = 625/64 ï‚» 10 1.25 is close to the answer.
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This also means that unknown^10 = 10 1^10 = 1 and 2^10 = 1024, thus answer is between 1 and 2, and probably closer to 1 than to 2 because 10 is closer to 1 than 1024. So we know that answer equals 1 plus a fraction (which is probably less than ½). 1.1^2 = 1.21 1.1^4 = (1.1^2)^2 = 1.21^2 = 1.4641 which is ~ sqrt(2) so 1.1^8 = (1.1^4)^2 ~ (sqrt(2))^2 ~ 2 and 1.1^10 = 1.1^8 × 1.1^2 ~ 2 × 1.21 ~ 2½ so 1.1 is too low. 1.2^2 = 1.44 which is ~ sqrt(2) so 1.2^4 = (1.2^2)^2 ~ (sqrt(2))^2 ~ 2 and 1.2^8 = (1.2^4)^2 ~ 2^2 = 4 1.2^10 = 1.2^8 × 1.2^2 ~ 4 × 1.44 = 5.76 so 1.2 is also too low. 1.3^2 = 1.69 ~ 1.7 1.3^4 = (1.3^2)^2 ~ 1.7^2 = 2.89 ~ 2.9 1.3^8 = (1.3^4)^2 ~ 2.9^2 = 8.41 ~ 8.4 1.3^10 = 1.3^8 × 1.3^2 ~ 8.4 × 1.7 ~ 8 × 2 ~ 16 so 1.3 is too high. 1.25 = 5/4 1.25^2 = (5/4)^2 = 25/16 1.25^4 = (1.25^2)^2 = (25/16)^2 = 625/256 ~ 2.5 = 5/2 1.25^8 = (1.25^4)^2 ~ 2.5^2 = (5/2)^2 = 25/4 = 6.25 1.25^10 = 1.25^8 × 1.25^2 ~ (25/4) × (25/16) = 625/64 ~ 10 1.25 is close to the answer.
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