วีดีโอ

No Calculator Allowed....proceeds to instruct the viewer to type it in a calculator. I'm disappointed.

= 1111*9999= 1111* 10000 - 1111*1 = 11108889

You spent more time doing all that when you could have just squared 729

Блин, ну зачем тат подробно? Да ну тебя, достал подробностями. Не буду больше тебя смотреть.

K^K=100 K≈3.5972850235404175054976522517822860691355430548865767837202521279729575075559739318180352240370591783302049767484398276551040776450252525589851913632736498774688621277871392188912176746980172077258791030612994167552954822581292156900947871971993555313663226231092003715474996189066164280340083858070020673586505762347755749649700502744653453422706324548283476744097084045096798003202407988940652787090870473610598619950561104479650804998269139871571243627278937664567039058629214343255675258434547654723779734037647339986100908922007864901625424234639642851844790057809948110491996763429647308469394459231543119496108739317264597589575291620634109007481252831565462047242234138971444631428595572994963882319484710581055252775092055054236925800397438759772899237254285966827424646741430212859822435319568246418186030411318896138821763351809755874289453640645861579747607628301738777525918530717755581424178011232809111429157206427876243187855877725554074455231057942376696415064405507004553617670379441639020472248407578973741933808593526205214629774895036789438741574515471213703614755051421149690698387743983477023156225079962353792968142201030397152031066373873017570050739656805321511177585181640584959845794918133293397979655248991814988470165452832209898046606406634764234878734988578895276094476447821968624502292537825620594556622196578526016411245370920666601466069557110189520193259163580435807541749735297612709601639015903179376442551389357672359404107763852028646573186009057145061151324330241989924323639619511176358464859608207993580337212900011724584405099810078669086887289842631758274878562107022679723613496136871897306067326691595439173425398992709845102367246009772317068976790747996106632101350789066843183148233933499274163011123167572936085697986145302476583597001304141650040890967630966432318555005221692661786073423052633365172564980619312918812267150407572744833551326339808742903201935312277284017371093891141841281865733400200178945668697168880536459160923557516612361148778949947373798447442194231240856828634535882665510557776831268396282468226788577862087966986912668475469638652605172605782194974789811572324823957561357397559620101764329119632936083246114357942991172625285345359047145397877856184009511132449498862248545283731748318733033899047145453737614565208192550408675470453669481173060852784411819980936364498632366537989475544771223284040333028237048085515258860643016998300214225329720530081700169587851539896260666935871740074895984301659518379307844773840768131313137282918618074146929761587997507562068011980130307395506242665303530687705538927998624085255825889265600708812606491920372899648639899513744219913961252965046946879276183358905582377328065329687038891628647059161650923445781220167550794712191861411606745843670917212695039162893199287210409010358296030047342014202665992605243660228560710225994281959725950912648610296971818623245319798731004931416127521628838238221104388070905888555425094814216960082895550773777301225415201824287497448052859665546075947715934685863184700242841768671789076718783093248755661503640307447836912695815198673557999018330091357130075531820258213342799666700160515570612049346900712805658513368523579142600088618075498853256982691419472547712619424939998124717806465530643248540586140496922277910226790431509112561195306486306620361540895487549014799603144032532374850773822294006614923196294044207541141147898987828785096072396501534934042313361081471206665520144620974322177871325349625731998753380027465468270708352214193985642228774866062980692641271408477559747183013530672469338413026302278560754098906465614912085963453774834681639070289880128174214846702200337055121274553629272688909058242835959425251698570504789037253939023992625418328297150588184183221634157545961522130143887604277120045781101733287282061257273837896881558525110547253999987864667818777536145305778733363307970277740514655098545201938324070530263809012143466240250099292377366750295167735501627260357448299068278815368577416073745723127445568387700835549893844277312505188609305667472465609799779514903799145086628004820032785531236778496287318770627669934160737699829831354339257219347425077262748144927528041368333539367654353004804138833796145153555801770114457344466093589671843450767054062708409404867058128243463659739054442027131413887194209429140338838414503736745149958160564444381551084576585809904285759334940867830829652723715790495696099111046446052336804764413199259761769567146029650223235442560618138399376875376810367663853784785433030701702370061437119206088469360480616920854376431811630660228762127650119401595151243203584753912062624882011369854616790837854107525442769478359426383772334204112687053624144850965372415067241257000193664868218948976848952520411355804214131491945137695845863626325041093404267775412835227317344133498053643979953711921930148803866512322105615262702688310851028717076739830371298537752690074240890525513041024110833075498785547186092790721722108675654705309005318232781689718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3^4^2=3^8?

3^(-1/3)

9^900 - 9^901 = 9^900 (1 - 9) = -8(9^900)

For (k-3)^2 - 2^2, you expanded and then factorized, instead you should have applied the difference of squares rule once more to get (k-3)^2 - 2^2 = (k-3 + 2)(k-3 -2) = (k-1)(k-5)

(k - 3)^4 = 16 k - 3 = 16^(1/4) k - 3 = 2, -2, 2i, -2i k = 5, 1, 3 + 2i, 3 - 2i

Seems waaay too loooong. All you have to do is remember powers of 2 up to 2^10, viz. 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024 2^15 + 2^12 + 2^9 + 2^6 + 2^3 = 2^3 * (2^12 + 2^9 + 2^6 + 2^3 + 1) = 2^3 * (2^2 * 2^10 + 2^9 + 2^6 + 2^3 + 1) = 8 * (4 * 1024 + 512 + 64 + 8 + 1) = 8 * (4096 + 512 + 73) = 8 * (4096 + 585) = 8 * 4681 = 37448

You math artist

3^(-1/3)= 3^(1-1/3-1)=3^(2/3)*3^(-1) =3^(2/3)/3. This does not simplify anything. It merely states that (1/3) = 1-(2/3)

Nice but a bit obvious

3^13 isn't that hard to brute force, is it? Could just multiply 3^13 = 3^5 * 3^5 * 3^3 = 243 * 243 * 27. Shouldn't be slower than the length of this video.

Le résultat final est plus compliqué que l’énoncé !!!!!

What if I just simply multiply 125*625 to get 5^7 and take away 6? (125*625=625*100+625*20+625*5=62500+12500+3000+125 and so on...) Does not seem to be more work, than 124*126...

Antilog(13*log3)=10m-3

5^7-6= 5*5^6-5-1= 5*(5^6-1)-1= 5*((5^3)^2-1^2)-1= 5*(5^3+1)(5^3-1)-1= 5*126*124-1= 630*124-1= 630*100+630*20+630*4-1= 63000+12600+2520-1= 78120-1= 78119

^My solution: =5*5*5*5*5*5*5-6=25*25*5*25-6=625*5*25-6=625*(10/2)*25-6=(6250/2)*25-6=3125*25-6=3125*(100/4)-6=(312500/4)-6=((300000/4)+(12500/4))-6=75000+3125-6=78119 all in my head without writing anything down Much easier!

Recall the famous Phytagorean triplet (3,4,5) m²=333²+444²+555² =555²+555² =2(555²) m=555sqrt(2)

27+24+23 (2^7=27) #euler #huxley

Calculation part not impressive.

340^2-337^2-3^2= (340^2-3^2)-337^2= ((340+3)*(340-3))-337^2= 343*337-337^2= 337*(343-337)= 337*6= 2022 Seems, that i found another method ....

sqrt(500(501)(502)(503)+1) x=500 sqrt(x(x+1)(x+2)(x+3)+1)= sqrt(x(x^2+3x+2)(x+3)+1)= sqrt(x(x^3+6x^2+11x+6)+1)= sqrt(x^4+6x^3+11x^2+6x+1)= sqrt(x^4+4x^3+6x^2+4x+1+2x^3+5x^2+2x)= sqrt((x+1)^4+2x^3+5x^2+2x)= sqrt((x+1)^4+2x(x^2+2x+1)+x^2)= sqrt((x+1)^4+2x(x+1)^2+x^2)= sqrt(((x+1)^2+x)^2)= (x+1)^2+x= 250000+1000+1+500= 251501

3, 4, 5 right triangle scaled by a factor of 1111 means that by Pythagoras Theorem, the answer has to be 3333^2

There is an obvious solution n=2. n^4 - (n-2)**4 = 8n^3-24x^2+32x-16=0 Since n=2 is an obvious solution, and 8 is a common factor of the coefficients of the cubic equation divide by 8(n-1) to obtain n^2-2n+2=0 from which we have two complex solution 1+i and 1-i.

3:50 8•49•50+7= =400(50-1)+7=20000-400+7=19607

2^25-2^24-3^2= (2*2^24--2^24)-3^2= 2^24*(21)-3^2= 2^24-3^2= 2^(12*2)-3^2= (2^12)^2-3^2= 4096^2-3^2= (4096+3)*(4093)= (4100--1)*(4093)= 4100*4093-4093= (4000+100)*4093--4093= 4000*4093+100*4093-4093= 16372000+409300-4093= 16781300-4093= 16777207 This looks easier to me than your solution, but our mileage may vary. I would also tend to write 4096^2 as (4100-4)^2 instead of (4000+93)^2, if i woulld go te way of our solution. 4096^2=(4100-4)^2=4100^2-2*4*4100+16=4100^2-8*4100+16=16810000-32800+16=16777200+16

Unnecessary lengthened the problem by using substitute.

nice solution

2:30 4096²-3²=(4096-3)(4096+3)= =4093•4099=(4000+93)(4000+99)= =4000²+4000(93+99)+93•99= =4000²+4000(200-8)+93(100-1)= =16000000+800000-32000+9300-93= =16809300-32093=16777207 😁

7(7^4 + 7^3+7^2+7+1) = 7(7-1)(7^4+7^3+7^2+7+1)/(7-1) = 7(7^5+7^4+7^3+7^2+7-7^4-7^4-7^3-7^2-7-1)/(7-1) = 7(7^5-1)/6 = 7*16806/6 = 19607. Simplifying by multiplying and dividing by (7-1) is very effective for sums of consecutive powers of 7.

It is much simpler to not use "m" at 4:13. Simply start easier multiplication there.

What a shame! Why not a simple calculation of 5 ×5×5×5×5×5×5×5×5×5×5 - 1 will be done. It will come as 3125×15625 - 1= 48728125 - 1 = 48728124. That's all.

Or just write it in binary

Overworked .... 4096 squared is simply 4096x4096 which you could do in primary school

Consider x=a(a+1)(a+2)(a+3) =(a²+3a)(a²+3a+2) =(k-1)(k+1) =k²-1 where k=a²+3a+1 --> x+1=k² sqrt(x+1)=k =a²+3a+1 =a(a+3)+1 Letting a=500, sqrt(x+1) is the quantity we have to find. sqrt(x+1)=500(503)+1 =251501

I got the answer correct.

I didn’t use a calculator.

2^25-2^24-3^2=16777207

4:09 Is there a difference between, x²+3x+1 (x²+3x+1) and (x²+3x+1)(x²+3x+1)? (x²+3x+1)(x²+3x+1)=(x²+3x+1)² x²+3x+1(x²+3x+1)=...?

9⁹⁰⁰(1 - 9) = -8¹9⁹⁰⁰ = -2³9⁹⁰⁰ = -(2¹9³⁰⁰)³

Much easier is to write 148 in binary and read the powers of 2 out. To do this choose the largest power of 2 that is less than or equal to the number you are trying to get to 148 = 128 + 16 + 4 = 2^7 + 2^4 + 2^2 = 2^2 + 2^4 + 2^7 = 2^a + 2^b + 2^c and there you have your answer: (a, b, c) = (2, 4, 7)

you can simplify the calculations by a lot by using (a+b)(a-b) formula, by choosing x the middle of all those numbers, i.e. x = 501.5 answer = a = sqrt(500 * 501 * 502 * 503 + 1) a^2 = (x - 3/2) (x - 1/2) (x + 1/2) (x + 3/2) + 1 = (x^2 - 9/4) (x^2 - 1/4) + 1 use y = x^2 to get a^2 = y^2 - (10/4) y + 9/16 + 1 a^2 = (16 y^2 - 40 y + 25) / 16 a^2 = (4 y - 5)^2 / 4^2 a^2 = ((4 y - 5) / 4)^2 a = (4 y - 5) / 4 a = (4 x^2 - 5) / 4 a = ((2 x)^2 - 5) / 4 a = ((2 * 501.5)^2 - 5) / 4 a = (1003^2 - 5) / 4 a = (1000000 + 6000 + 9 - 5) / 4 a = 250000 + 1500 + 1 a = 251501

Waaaaay tooooo loooooooong You can do it in your mind. Definitely not an Olympiads level question m^2 = 333^2 + 444^2 + 555^2 take the 111^2 common to get m^2 = 111^2 * ((3^2 + 4^2) + 5^2) Now we know the famous Pythagorean triangle 3, 4, 5, so 3^2 + 4^2 = 5^2 and 5^2 + 5^2 = 2 * 5^2 Now take square root of both sides, to get m = 111 * 5 * sqrt(2) = 555 sqrt(2)

I mean sure, it issolved that way, but what's beautiful about it? You are given two big numbers, you substract and get one number, that's still big.

We can assue a<=b<=c (otherwise, we can rename the varabes, so that this is given) Now we can tae out 2^a out of the sum: 2^a*(1+2^(b-a)+2^(c-a))=148=2^2*37 Since a<0b e hae the possibiities a=b and a<b. et us first oo at thhe case a=b: 2^a*(1+2^0+2^(c-a))=2^2*37 2^a*2*(1+2^(c-a-1))=2^1*2*37 Comparison of the exponents of 2 lead us to a=1 and therefor 1+2^(c-a-1)=37 2^(c-a--1)=36 But 36 is not a power of 2, so a=1 is possible (for whhole nummbers a, b and c). So we now a<b<=c and 2^ a*(1+2^(b-a)+2^(c--a)=2^2*37 (1+2^ (b-a)+2^(ca)) is an odd number, so compaisoof exponents of 2 on both sides gives us a=2 and 2^(b-2)+2^(c-2)=37--1=36=2^2*9 Now we can use the same trick:.. 2^(b-c)*(1+2^(c-2-(b-2)))=2^(b-c)*(1+2^(c-b)=2^2*9 Again we oo at thhe 2 cases b=c and b<c. The first gives us no solution, so we get b<c and 2^(b-2)*(1+2^(c-b))=2^2*9 Since 1+2^(c-b)is odd, we get 2^b-2)=2^2 and 1+2^(c-b)=9 b-2=2 and 2^(c-b)=8=2^3 b=4 and c-b=3 b=2 and c=7 So the complete solution is a=2, b=4 and c=7. Chec results in 2^a+2^b+2^ c=2^2+2^4+2^7=4+16+128=148 Why do you rewrite 2^(b-a) to (2^b)/(2^a)? it is unneccesarry. Just tae out 2^(b-a) outt of the su, and ou get 2^(b-a)*(1+2^((c-a)--(b-a)))=36 The first factor is a power of 2, the second factor is odd, so we get 2^(b-a)*(1+2^(c-b)=2^2*9 and than 2^(b-a)=2^2 and 1+2^(c-b)=9 b-a=2 and 2^(c-b)=9-1=8=2^3 b=a+2 and c=b+3 Togetthher withh a=2, we have a=2, b=2+2=4 and c=b+3=4+3=7

I did it in my head.

Input sqrt(500×501×502×503 + 1) = 251501 Result True 251501