Neil deGrasse Tyson Explains Exponentials

Why are humans so bad at thinking about big numbers? On this explainer, Neil deGrasse Tyson and comic co-host Chuck Nice wrap their heads around large numbers. It’s exponentially confusing.
Find out about linear versus sequential thinking and how we approach increments. How do we get better at thinking exponentially? Discover why models of the solar system are inaccurate. How many earths can fit into the sun? How many suns can fit into the red giant version of itself? Also find out why Chuck needs to call a mortgage broker. All that and other reminders of why the universe is under no obligation to make sense to you!
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About StarTalk:
Science meets pop culture on StarTalk! Astrophysicist & Hayden Planetarium director Neil deGrasse Tyson, his comic co-hosts, guest celebrities & scientists discuss astronomy, physics, and everything else about life in the universe. Keep Looking Up!
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[@schizzzyjones9966]

I absolutely love these guys together! This is a rare friendship between comedian and astrophysicist. They always make my day and make learning THAT much better!!

[@schizzzyjones9966]

Neils laugh is that of a child’s laugh! It’s so genuine and I mean that in the BEST way possible guys!!

[@lilnastyy1604]

It’s the beauty of intellect in different form. Comedy is a science in itself ✨

[@schizzzyjones9966]

@@lilnastyy1604 yea for sure I agree 💯

[@mangledmango]

100% 1/1

[@C-LOS420]

I agree, I watch daily only because I need some normalcy with people who have common sense 😂

[@M_Alexander]

Chuck really deserves more credit for just how well he actually learns everything they talk about. He might be confused at first but a few months later he applies it to something even harder!

[@alafia1024]

His enthusiasm and understanding is contagious.

[@joshuaperrine2019]

Chuck is so funny! I love the blend of humor and science from this series.

[@boboloko]

He seems really humble and down to earth

[@mookibois9784]

c a p

[@SAJe_53]

None of my teachers seemed to be of that opinion.

[@jesusceferinopatino9430]

Another exponential I like is time in seconds. We understand 1 second, 10 seconds and 100 seconds. 1,000 seconds is 16.67 minutes. 1,000,000 (million) seconds is 11.6 days. 1,000,000,000 (billion) seconds is 31.7 years. 1,000,000,000,000 (trillion) seconds is 317 centuries.

[@linyenchin6773]

@@bru-rz1pi number of seconds, not "numbers of seconds," because; *grammar.*

[@randallmcgrath9345]

This may be dipping into number theory, then.

[@SherubThakur]

I like to convert other exponentials to time. It just helps me understand stuff better. Like we can (arguably) appreciate how much is 1second, to 1 year, or even 50 years, (above that it becomes hard IMO). But it is a great metric to understand other exponentials. Let's take the example of packing earths in a hollow sun: if we could pack 1000 earths per second - it will take 16.67 minutes to fill up the sun - it will take 31.7 years to fill up the sun (when it is a red giant) Another fun one is, understanding the wealth of an individual - What if you spent $1 per second how much time will it take you to go through $1M or $1B. - 11.6 days for a millionaire - 31.7 years for a billionare It just helps one to understand just how much difference is there between the wealth of a normal person and a billionaire.

[@Nauda999]

And let's assume our Universe is infinite, and let's start thinking about different infinite universes.

[@floggyWM1]

if a human body has 7,000,000,000,000,,000,000,000,000,000 atoms, how many atoms are in your city.

[@I-VisiBomb-I]

11:24 Neil dropping the HARD F when it comes to banks

[@sweetdurt2143]

My name is Yoshikage Kira. I'm 33 years old. My house is in the northeast section of Morioh, where all the villas are, and I am not married. I work as an employee for the Kame Yu department stores, and I get home every day by 8 PM at the latest. I don't smoke, but I occasionally drink. I'm in bed by 11 PM, and make sure I get eight hours of sleep, no matter what. After having a glass of warm milk and doing about twenty minutes of stretches before going to bed, I usually have no problems sleeping until morning. Just like a baby, I wake up without any fatigue or stress in the morning. I was told there were no issues at my last check-up. I'm trying to explain that I'm a person who wishes to live a very quiet life. I take care not to trouble myself with any enemies, like winning and losing, that would cause me to lose sleep at night. That is how I deal with society, and I know that is what brings me happiness. Although, if I were to fight I wouldn't lose to anyone.

[@reversethepolarityoftheneu773]

HAYATO

[@sweetdurt2143]

@@reversethepolarityoftheneu773 ah I see, you are a man of culture as well

[@robertnobles8189]

Jesus preached Peace until he got to the money lenders. F them!

[@linyenchin6773]

An unfortunately effeminate move on his part; drowning in emotion, men don't do that.

[@alanbrady420]

Neil: “Chuck I’m going to tell you how your brain is wired” Chuck: poorly 🤣🤣 Two absolute legends, I love watching and learning from your uploads, and the added comedy is class!👌🏻 keep them coming.

[@mattevans-koch9353]

I agree. Where was this fun, humorous education when I was in school fifty plus years ago? Love these little mini classes.

[@tuneboyz5634]

@@mattevans-koch9353 have fun Matt. Enjoy the mini classes :)

[@SAJe_53]

Yes, Alan. We all heard.

[@alanbrady420]

@@SAJe_53 oh.

[@davidt3956]

Most homeowners had no idea of the mortgage risk. However, all the bankers had the education to know what they were doing. Homeowners should have been protects and the bankers should have been allowed to fail.

[@Grimloxz]

EXACTLY. I don’t know how many times I e tried to explain this.

[@finnracing99]

And the bankers are still getting away with legal swindling to this day. Until the majority of peeps can figure out how to steer away from instant gratification, the bankers will continue to get these crazy interest rate profits. The bankers really know how to take advantage of the masses in order to make their profits and they have zero compassion for the everyday person just trying to survive. Moral of the story: if you can avoid borrowing money from an institution, then the long needed change of power can be switched back to the everyday people. Just my opinion. This is easier said than done obviously

[@tokonjudo]

'In ten days it will cover the earth', made my evening great, cheers chuck.

[@Fister_of_Muppets]

"In ten days it will cover the Earth" - Ron Jeremy, 1994

[@alexandercanella4479]

Also its accurate if you consider the average size of a lake 😂

[@mattevans-koch9353]

Is it possible that our innate linear thinking is what hampers our ability to grasp the spread of things like viral outbreaks? The penny doubled everyday is very similar to the speed of disease spread especially with modern travel capabilities. Thank you Neil and Chuck.

[@agnosticignostic4086]

Absolutely, and also the fact that acting only a day later or a day sooner can have dramatic effect on the long term

[@alexandercanella4479]

That's the entire purpose of R-naught. I would have used the 0 for naught but that's not an option on my phone lol.

[@emmanuelmunkondia3866]

That's why we trust math more than our own opinions

[@robertnobles8189]

NDT “Chuck, I can hand you 5 million dollars right now” Chuck “I always knew I liked you!” 😂🤣

[@Fastwalker27]

There is a VFX video made by the corridor crew , which uses CGI to portray accurate sizes of astronomical objects to help understand the scale better , I highly recommend watching it

[@chrisbarnard1863]

Their videos are pretty sick all round

[@ismailkoca2307]

1 is making the science, space subjects more understandable, the other 1 is making the video funnier. And both of them are like legends about that. I saw StarTalk youtube channel just a few days ago, but i watched many of it's videos. Thanks Mr. Tyson and Mr. Nice for making these subjects funnier and more understandable :)

[@jarrodfrankum]

Chuck totally unlocks Neil with the humor and really opens the conversation. You can see Neil explaining and Chucks gears turning. This is such a special relationship

[@GrimJerr]

“Space is big. You just won't believe how vastly, hugely, mind-bogglingly big it is. I mean, you may think it's a long way down the road to the chemist's, but that's just peanuts to space.”

[@GrimJerr]

Douglas Adams

[@rajdeepsingh1019]

Combination of Dr. Tyson and Chuck is like H2O, very crucial for survival of my sad soul. Much thanks.

[@waywardgoddess7219]

I can't stop binging these StarTalk videos and I just bought your book lol so I know the line you were talking about. Absolutely love you guys!

[@reversethepolarityoftheneu773]

14:17 this has to be my favourite quote of all time it perfectly sums up why science is so hard the world is not made to be easily understood by us and we are not made to understand it.

[@ktrishan3165]

mine too...after reading the book

[@msmith53]

Yet.....!

[@robertnobles8189]

And then try quantum mechanics! 🤯

[@magellancharteringsolution5288]

Great video as usual. Please make a video describing all earth’s movements and their cycles (incl. rotation, revolution, precession, Nutation, around barycenter sun/moon etc). Thanks

[@philipcollier4883]

In Chuck's defense, he was probably taking into account real world conditions where the growth rate of the algae may have slowed due to algae dying from overcrowding

[@Heraclitean]

Probably.

[@acepilot1]

I wish this video was exponentially longer, I could listen two these two for the rest of my life

[@brandtreppond2167]

I'm so happy I can hear you talk about the things you know

[@IsaacPiera]

Exponentials grow very fast. Factorials: Hold my beer.

[@nigeltufnel4031]

1.2 e+33

[@StaticBlaster]

Try tetration. (repeated exponentiation)

[@StaticBlaster]

and knuth's up arrow notation.

[@tidus9942]

@@StaticBlaster what I said. Arrow notation is basically increasing an exponent amount exponentially. First you have something to the power of 3 and then the power of 27 and then add in the next arrow and it gets to unfatherly large numbers, lol.

[@Merto6]

tree function: lol noobs

[@literallyh3093]

I love that were diving into this topic again.

[@Jasems001]

I just found this channel last week. You guys, the content, and the way it is delivered is amazing.

[@dracoargentum9783]

Oh, you'll have a blast going through archived content!

[@gnarthdarkanen7464]

A handful of words on "scale"... AND of course, I love these two!!! They're always fun and informative... When I have trouble dealing with the scale of a problem... depending on the particular problem, I try to visualize something more relatable to connect to it... Scale of the earth for instance... and then the sun... If you Scroogle it, the earth is around (not exact) 10,000 miles in diameter, which when one considers C= pi * D... comes to 30,000 miles (and some change) around... I consider the old-school rule to change oil every 3,000 miles or 3 months (of regular driving)... SO to drive straight through the earth, I'd change the oil 3 times and still have about 1000 miles to go... my state is around half that from where I live to Memphis (on the Miss... River)... SO in the "regular" habit of driving I do, it would take me about 10 months to make that journey... through... AND 31 or 32 months (2 and a half YEARS... or a tad more) to drive around it... Now... Take a dime and use it to draw a circle on a wall (or a paper you then tape to the wall... Let that dime-size circle represent Earth... If in the same "picture", the circle that would represent the sun would take up THE ENTIRE WALL from floor to ceiling... Note, please, that it's still NOT a perfect scale for precision, but you're now able to visualize (give or take) WELL within the "ball-park" of what you should think of in the comparison... AND a little closer to how it might connect or relate to your own life... Struggle with the millions, billions... trillions? Okay... A million dollars in $100 bills, is about 10,000 actual bills and bundled together, it weighs about 5 lbs... A billion would weigh 5000 lbs in $100's... So you couldn't carry it all in the bed of a full-ton dually... You'd need 2 duallies AND a half-ton pick-up to legitimately carry a billion dollars in $100's... Think about that... If you earned a dollar every second, you'd make 60 bucks a minute and 3600 every hour... It would still take you 32 YEARS to earn a billion dollars, and you STILL (at 3600/hour) would NOT LIVE LONG ENOUGH to earn a single trillion dollars... A trillion dollars would weigh about 5 million pounds in $100's... SO you'd need a FLEET of semi-trailers to load that cash and move it anywhere. Like I said, this ain't perfect or exactly precise. It's being able to "ball-park it" and find it relatable to something you CAN understand. ANYWAYS... I hope this helps someone... It takes a little repetition and practice, but it can be a useful practice guide. ;o)

[@shubhsrivastava4417]

There is an old story in India where an old sage challenges a king in South India in a game of chess. If the king won the sage would have to give everything he had but if the sage won the king would have to give grains of rice starting from one, doubling everyday till 64 days(because 64 squares on a chess board). The king thought linearly and accepted the challenge. The sage won. Till the first 15 days the king kept on giving rice and boasted about his kingdom's wealth but on the 20th day the king realizes that he has been tricked and asks the sage to forgive him. The sage forgives him and advices the king to think carefully before doing anything. Great story on exponentials!

[@michelerossi7121]

Always a pleasure watching star talk 👍🏼

[@Anti-HyperLink]

I listened to your audio book of Astrophysics for People in a Hurry. I wasn't in a hurry, I was chilling at home. I don't remember the opening line, but I once knew it.

[@StaticBlaster]

Neil should talk about Knuth's Up-arrow Notation and extremely large transfinite numbers.

[@sagarkardak7256]

Clock : 3 am Me : Sleeping NDT : BIIIIIIIIIIIIIIIIIIIIIIIIIIIIG NUMBERS....

[@klaxoncow]

"Mortgage" literally translates as "death pledge". In fairness, the original Latin "death pledge" meant that the debt would either terminate with your death or eventually be paid off. So it's not really as bad as the name "death pledge" makes it sound. But, still, as you pay off your mortgage, remember that "mort" means "death" and you're paying off your "death pledge". And, yeah, paying that mortgage probably will be the death of you too, eh?

[@Geezer-yf8hv]

Sounds about right to me, according to my experience!

[@harryscarry6064]

Wow. Makes sense. Think I’ll be dead before before my death deal is finalised.

[@denisemcdougal6445]

Wow !

[@tenoki]

Another Latin nerd in the house.

[@wisewina]

That's a sweet and calming statement though....there is something called 'death insurance'....actually I meant 'Life insurance', the fact is that if you die, the bank would be paid anyways.....the banks are affiliated with insurance companies, and the bank itself can insure you through other insurance companies....my point is, the phrase "death pledge" regardless of what it meant in Latin, it's in some manor factual in essence,

[@KK-hp5bi]

1:28 Swahili: Kimeumana! If you know you know.

[@roddan404]

Accurate 💯

[@winnieroze]

😄

[@shakesrear7850]

Thanx guys. Massively appreciated

[@strider0303]

I absolutely LOVE these explainer videos

[@DetroitDave76]

Love that Mr.Tyson and Chuck do these videos ! watch these videos learn and laugh !

[@joanhall3718]

I hit the thumbs up before I even watch the videos. Every one is entertaining, informative, and just simply a delight.

[@adrianscarlett]

It amazes me that a number as large as the quantity of atoms in the universe can be represented by the term 10^80

[@carbrickscity]

Math is way more powerful than people think. Something as short as 10^10^100 is far beyond almost anything in the known universe. You can easily create a notation is far far beyond that as well.

[@MICKEYISLOWD]

That is the observable universe. We know it has to be 250 times larger at least because it is so flat. It may be it is exquisitely flat and is infinite. It boggles my mind that the universe could be infinite when it was smaller than a proton and still infinite now it has stretched out as we see it today.

[@teresaamanfu7408]

Great stuff!👌🏾😊

[@maddmann]

Love u chuck been listening to start talk for years first time seeing u

[@MaegnasMw]

We all knew Dr. Tyson was brilliant as a scientist AND as a communicator of science to the general public. What is surprising, pleasantly, is how sublimely clever Chuck is, in his grasping of subject matters far removed from the everyday experience or the average "layman" level of understanding. He may play the fool, so to speak, for comedic effect but he is anything but. A worthy sidekick, a great comedian and really fun to watch him listen, learn and react. These two are the best science teachers a man could ask for!

[@owlthemolfar4690]

Thank you for the video! Now whan Chuck is almost ready, next big number theme is Knuth's up-arrow notation and Graham's number? To proove we can create integers without any phisical meaning.

[@TRKPurgatos]

Been here from 100k subs I love that this channel is growing

[@imsoemo2234]

These are so much fun to watch. And I learn things too, I can tell the guy's are just having a good time too, maybe that's why this is so great

[@mikebeemd7296]

Man I love these videos, learning so much

[@tatotato85]

I thought i had a handle on exponentials but then i was presented to arrows notation and it blew my mind with things like Grahams number

[@projectlifedog1459]

Every Vsauce opening: Hey Vsauce Michael here! Startalk: Chuuck

[@howardOKC]

LOL love the Chuck

[@gdavies4]

My two favourite providers of astrophysics on KZbin 😊

[@cyndicorinne]

Wow this is deep. Especially the discussion of floating interest rates and the 2008 collapse.

[@MaryAnnNytowl]

Yes, no one should take a loan unless it has a fixed interest rate. Ever. You can always refinance if the interest rates drop, and get a better fixed rate!

[@jsharp5680]

Many of us who have engineered computer hardware or software understand these concepts, especially if you've started around the birth of the personal computer age in the late 70s / early 80's. Dealing with storage, throughput / bandwidth, random access memory capacities over the years absolutely solidifies an understanding of exponents and exponential growth.

[@ryanbensberg5846]

The people who you claim cant think in terms of long term in regards to evolution are typically Religious, Ironically are the same people who think in terms of eternity. other than that, I love you two and your videos.

[@stephenjeffrey3344]

Absolutely brilliant!!

[@minimalgrammar1276]

"Everything's big in astrophysics. Texas, you lose." I love the energy of this line.

[@michaelroddy6559]

If you cut Alaska in half, not only is Texas the 3rd largest state in the union; it's not that close to the second 😂

[@tangyenebanehsejine9330]

ahhh... my daily coffee. Loooooooooove watchx u two talk. I love your chemistry :)

[@JonathanSeni]

Wow, this is amazing, habari from Tanzania, where Serengeti is located.

[@RealCyclops]

13:48 - This also explains why people refuse to believe that the foods we eat already are GMO, but it took thousands of years to perfect the current strains. The only reason people are concerned now is because we can see a change within a decade now, not 10 lifetimes.

[@tidus9942]

I really want to see neil talk about arrow notation which makes number increase at an even greater rate is size. Is still using exponents but the exponents themselves grow exponentially.

[@pjg6019]

Love your chemistry

[@UnsealedGames]

Best podcast ever! 👏

[@Starman141]

love this! you guys are hilarious and educational!

[@suefalls1600]

Great info! Space uncomprehensible!!

[@00Skyfox]

That “start with a penny and double it” thing is an idea I had years ago for a game show. Similar to Jeopardy!, it has 30 questions with the first right answer awarding 1 cent, then every question after that will double the amount if right, or halve it if wrong. If someone gets all questions right, they get a total over $5 million (or add a 31st bonus question to bring it up to $10.7 million).

[@charonrose3306]

You two are a great team 💙😗

[@SuV33358]

I love these guys together. I learn so much, and laugh too! I wish I had Neil for a teacher

[@crazyboy74100]

that is a great observation about the models of the solar system and how they exaggerate the planets

[@Justin-gv3lp]

I was thinking about this the other day. I read somewhere that the largest black holes will eventually evaporate after something like 10^100 years from now. Which means after 10^99 years, that unfathomable amount of time has to elapse again another ten times over before you get to 10^100. It’s impossible to visualize how much time that is

[@onborders9398]

Chuck with a new mic, Nice!

[@lkbigfish1]

Chuck bringing heat right off the bat! Great vid.

[@zealest2011]

This is so cool!

[@DagKlingstedt]

Educational and funny - the best combination ever!

[@heyitsimme2335]

so very true. thank you for this!

[@davidevans3227]

yeah i can't get my head round how big the sun is too.. thankyou for making and sharing these great videos 🙂 x

[@benttranberg2690]

I watched the preview on mousing over, and thought "what kind of torture is going on here?" So now I'm watching. If only school was this fun! This reminds me when I was reading Asimov decades ago. Unfortunately he only authored around 500 books.

[@akshaaysharmas]

Love this duo

[@skonther0ck]

I’m frivolously but not exponentially really curious about that gold column behind Neil’s right shoulder.

[@dracoargentum9783]

I want to know about the Didgeridoo in his office...

[@bradfordcotta5925]

One of the great examples is when you ask someone how tall a piece of paper would be if you could fold it in half 50 times. Your brain simply can't fathom that the answer is "it would stretch to the sun" but it would. Fascinating.

[@tuneboyz5634]

wow really? Thats a crazy fact. Thank you for sharing!

[@denisemcdougal6445]

Thank you !!!!!!

[@callmeslick860]

I heard you talk about the 4th dimension and I would like to say . I had a thought and it was “ everything is already in existence it’s just yet to be discovered “ in other words a Tesla vehicle was already in existence because the materials were on earth already…. But Neil thanks for inspiring me

[@lakhyee7781]

Can u guys make one video on time crystals please 🥺🥺

[@ujjc001]

Hey Neil, can you do a video explaining what the maximum brightness that something could ever be?

[@vykintasmorkvenas6839]

Compared to what?

[@tuneboyz5634]

Vsauce made a video on that

[@jefgir]

@@vykintasmorkvenas6839 not sure if your reply makes sense

[@vykintasmorkvenas6839]

@@jefgir the original question doesn't make sense itself.

[@JOMFSE]

Our eyes only see a certain portion of light. You can see a small flashlight from light years away if there’s nothing to distort it. Hope this helps and hope I’m right haha

[@Vastin]

A few key exponents that hit us in real life on a regular basis: - interest rates - inflation - population growth - infection spread - social media disinfo spread

[@markusmcgee]

Thanks!

[@Lawsonomy1]

7:40 "As a astrophysicist we are steeped in it. We are Baptist in it. So we have a easier time." Not just Astrophysicist. As chemists we routinely deal with sizes, scales, and lengths of time that are ridiculously small. I remember earlier on in my training doing drills in exponentials. Converting back and forth between exponential notation to chemical concentration and back over and over. Now it's like my brain as a zoom function, can easily imagine zooming in to picture things that are super small, or zooming out to picture things that are super large.

[@donnad7426]

I speak to people about exponentials all of the time .. most recently when discussing population growth & now Spotted Lantern Fly growth / spread .. I think the lantern flies will give people a better sense of it.

[@avontmorganjr7505]

I met Chuck. He's a wonderful family man. He and Neil make a great team.

[@hantihut8141]

Hi Tyson, could Benfords Law be a subject you would take up sooner or later on your channel.......ex distances from the stars to earth etc. fits, and how does that and what does that say about the univerrs.........

[@aprilk3002]

I LOVE THIS SHOW!

[@vonkug]

I own your book Astrophysics For People in a Hurry in hardback, and that is not the opening line from the first chapter or the preface.

[@Seadalgo]

It would be great for you guys to discuss the ways that the human brain is wired for exponential perception, like light and brightness

[@CheddaWhizzy]

12:50 Exactly, bro. In fact, 1000 seconds is about 16.67 minutes whereas a million seconds is equivalent to about 12 days. And a billion seconds is equivalent to about 32 years. That's the power of exponentials.

[@BargedIn]

As much as these type of videos may make others feel small, they make me feel large, for I can witness them.

[@shadowprince4482]

As soon as he said Moorgate rate I knew I knew I needed an analogy.

[@denisemcdougal6445]

I love that The universe is under no obligation to make sense to you

[@grumblewoof4721]

A bit late, everyone in the world needed this at the start of the Pandemic.

[@dracoargentum9783]

if you buy from the links in this page, you're guaranteed to be supporting them.

[@jevicci]

Ironically, even though we THINK linearly, we SENSE things exponentially (or logarithmically). As an example, the difference in volume between a 1000 microwatt and a 2000 microwatt sound is not perceived as the same as the difference between a 2000 microwatt and 3000 microwatt sound. Instead, it takes a 4000 microwatt sound to be perceived as an equal increase in volume, and an 8000 microwatt sound for the next proportionate increase. So, instead of perceiving linearly (1000, 2000, 3000, 4000, etc.), we perceive logarithmically (1000 x 2º, 1000 x 2¹, 1000 x 2², 1000 x 2³, etc...). In this way, sense organs are useful over an enormous range of intensity.

[@msmart2u2]

Best show on the internets!

[@Kevin19HDS]

Love Neil and Chuck

[@bulukacarlos3571]

An excellent example that a professor of trigonometry gave us (who from Argentina communicated by mail with Carl Sagan several times): Suppose that a newspaper sheet is 1 millimeter thick, fold the newspaper 50 times ... you will reach the belt of asteroids !!!, Greetings from Argentina. Un excelente ejemplo que nos dió un profesor de trigonometría (que desde Argentina, se comunicó por correo con Carl Sagan varias veces): Supongan que una hoja de perióico mide 1 milimetro de espesor, plieguen el diario 50 veces... llegarán al cinturon de asteroides!!!, Saludos desde Argentina

[@johnstrickland3373]

Chuck plays the dumb sidekick but he is no slouch mentally and brings so much comedic relief to all this lofty mathematics, I love these guys together

[@otayplayz7617]

7:45 I think I can say for everyone that we all had a math problem like this in school

[@georgestill2473]

Amazing when the brain make you go from ultra instinct, to slowing down time, then making you go rag doll mode all within seconds 😁

[@reidflemingworldstoughestm1394]

Arithmetic starts getting weird at exponents. Counting, the most basic arithmetic, makes perfects sense -- just add one at every new step to what you had at the last step. Adding is a lot like counting, but you can do it in bunches: start with 5, glom 3 to it and you've got 8. And multiplying is just a ramped up form of addition. If you have 5 sets of 3, you're basically adding by 3 -- 3, 6, 9, 12, etc -- 5 of which gets you to 15. What these operators have in common so far, is that both numbers, 5 and 3, show up in the answer. That 3 is still sitting there next to the 5 inside the 8 you got after you added them together. Same goes for multiplying -- there's still 3 5s (as well as 5 3s) right there in the 15 you got when you did times. But now we come to exponents where, still working with 5 and 3, we would get 5^3, which means 3 copies of 5 multiplied together, so 5x5x5 which is 125. This is how exponents aren't like the other bois and girls -- there's no 3 in 125. Where did it go? 125 can only be divided by 5. It's like there never really was a 3. Really, the 3 is outside of the value, just working as a note off to the side to let you know how many of the numbers that you _are_ working with need to go into play.