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Huh? If you use your calculator at 3:35 anyway, why don't you use it right from the start and simply use to it calculate the log of 80 to the base 8?!?
I find this an unsatisfactory dive into the unknown. I like 1:55 but my instinct would be to take logs base 2 and then have: K = 32*L2(K) Clearly K is a multiple of 32 and inspection quickly leads to the answer. No need for all that garbage Lambert functional complication in other posts.
Not too bad. You knew m had to be a number between 2 and 3 , and8^2=64 and 80 is closer to 64 than 8^3=512, so you knew that m had to be a number just a little higher than 2. You could have solved it with trial and error but why bother when you can use logs. Have a great day!
exactly same until 5555^2-4444^2=9999 x 1111 after this =9999 x (1000+100+10+1)=9999000+999900+99990+9999=11108889 since no calculator allowed this is much easier to calculate, me thinks.
Anything a b both larger than e - put the bigger number in the exponent - here 16 ^ 18 is more than (16/e)^2 larger than 18^16 - note [(1+1/8)^8]<e - so you get (18/16)^16 <e^2 which is clearly less than 16^2. But the above just finds a convenient base, which works - but if you know (1+1/n)^n increasing for all n and lim n to inf - e you can show any combination “easily” and know which is more by inspection.
FWIW, I would have noticed that 3 is a solution by inspection (or graphing :D) and then divided k^3 + k - 30 by (k -3) which gets you to the same place. I don't think there's really a difference between noticing the solution of 3 and you splitting the number up. As a tip for your channel, I don't think you should spend as much time on the trivial steps (adding 9 + 1 to get 10...)
you all smarty pant of the comment section pls shut up, u never understand us idiots. this method is def longer, but everything makes sense so we can follow
Hadn't seen that fractional method, interesting. Just used the common denominator method. After reducing the resulting fraction, we end up at the same place with k / 15 = 150. k = 2250.
Another approach: recognize it is a geometric series and apply the formula for summing a geometric series. That gives (9^6 - 9)/8. Recognize that 9^6 and 9 are both squares and factor 9^6 - 9 = (9^3 + 3) (9^3 - 3). It is easy to do 9^3 = 729 in your head. Substituting gives 9^6 - 9 = 732 x 726. The answer is that divided by 8. We can do the divide by 8 before multiplying out 732 x 726. Divide 732 by 4 and 726 by 2: 732/4 = 183, and 726/2 = 363. Recognize that 363 = 121 x 3, and that 121 = 11^2. The final answer then will be 183 x 11 x 11 x 3. Multiplying by 11 is easy. Just add the number to the same number shift left one place. 183 x 11 = 2013. 2013 x 11 = 22143. 22143 x 3 = 66429.
If you are going to use paper and pencil methods of calculating the sum (which is not needed), you could just do the multiplication that way 81×82. Or in just head, its not that hard.
Yes. Seeing that 9+1=10 is indeed a good call. I would solve it by seeing that 9^2+9=90: So the original question becomes 9^3(9^2+9)+9(9^2+9)+9, or 90*9*(9^2+1)+9, which is 90*82*9+9, or 82*81*10+9, from this point, the solution is in the video.
It seems you randomly raised the two expressions to 1/11! Was it? You needed to explain why you did that as the solution hinges on that choice. Don't you think?
