Introduction: Take Cube Roots of 6-Digit Numbers Mentally

Picture of Take Cube Roots of 6-Digit Numbers Mentally

First Instructable, hope you like it.

Math. Love it or hate it, everyone must do it at some point in their lives. Some of us revel in numbers and equations and have a passion for math. This Instructable is for you, oh math lovers of the world.

This Instructable describes mental cube-root extraction: a cool math trick you can do to amuse and amaze your friends, and score some mean cubicle-cred with your fellow geeks. I can accept no responsibility for the social implications of using this skill (but then, as a geek, you probably already treat social implications of your geekiness with abandon, so it's all good).

At any rate, let's move on!

Update: Featured?!? Thank you so much Instructables, that means alot to me :-)
Update2: I know it's alot to ask from you folks, but please, vote for this in the book contest if you think it's cool. I would really appreciate that.

Step 1: A Note on Limitations

The technique I describe in this Instructable has a few important limitations that you should be aware of.

So, here they are:
1) This trick only works for perfect cubes, it will not work for any arbitrary 6-digit number
2) The cubed number must be an integer (whole number) between 0 and 100 (or 0 and -100). No fractions, no decimals.

For ways to expand the technique, see the last step of the instructable.

Step 2: Cubes/Cube Roots

Picture of Cubes/Cube Roots

In mathematics, a cube is any number raised to the third power. Said another way, it's any number that is multiplied by itself 2 times. For example 100 cubed = 1003 = 100*100*100 = 1,000,000.

The cube root of a number is exactly the opposite. Think of it as, given a number, the cube root of the number is the number you would have to multiply by itself 3 times to get the current number. So, from the above example:
Cube root of 1000000 = (1000000)(1/3) = 100

By the time you have finished this instructable, you will be able to determine the cube root of any number between 0 and 99 cubed (so, anything from 0 - 970299). And the best part is, you'll be able to do it without the use of a calculator, using nothing more than your mind.

Step 3: What You Have to Know

In order to extract all of these cube roots, you will need to have a small subset of cubes memorized. So, in order for this to work well, you must memorize these 10 cube roots:

03 = 0
13 = 1
23 = 8
33 = 27
43 = 64
53 = 125
63 = 216
73 = 343
83 = 512
93 = 729

Write these down on a sheet of paper and take the time to memorize these, it will make the upcoming steps much easier.

Step 4: Splitting the Cube in 2

Now, suppose we're given an arbitrary number that we know to be a perfect cube. For example, 50653 or 262144. We will tackle the cube root extraction by splitting the number into 2 halves, based on the comma separating the hundreds and thousands positions of the number, and evaluate those 2 numbers separately.

So, if our number is 50653 from above, we would split it at the comma, yielding 2 numbers (50 and 653) which we will evaluate separately, and then put back together to create the final cube root.

Next, we will evaluate the right side of the expression, yielding us the 1's digit of our extracted cube root.

Step 5: Evaluating the Cube Part 1: Right Hand Side

From the previous step, we took our number (50,653) and split it at the comma. In this step, we will evaluate the right hand side of the cube (653) to extract the 1's digit of our cube root.

The key to determining the 1's digit of the cube root is to look at the rightmost digit of the cube and compare with the rightmost digits from our list of cubes. The one's digit will then be the number from our list of cubes that matches the one's digit of the cube. So, for 653 we will want the number who's cube also ends in a 3. Looking down our list of cubes:

03 = 0
13 = 1
23 = 8
33 = 27
43 = 64
53 = 125
63 = 216
73 = 343
83 = 512
93 = 729

From above, we see that the cube of 7 also ends in a three, therefore, we can deduce that the cube root's 1's digit must be 7.

Now, we shall evaluate the left hand side of the cube, which we will then use to determine the number's cube root.

Step 6: Evaluating the Cube Root Part 2: Left Hand Side

From step 4, we split the cube root into 2 parts, evaluating both sides of the expression to determine the cube root. We already evaluated the right-hand side of the expression, now it's time to evaluate the left hand side. Using our example, we split the number 50,653 into two parts: 50 and 653. In this step, we will evaluate the 50, yielding the 10's digit of our extracted cube root.

To determine the 10's digit of our cube root, we take our left hand side (50) and pick the 2 cubes that it falls between. From our list of cube roots:

03 = 0
13 = 1
23 = 8
33 = 27
43 = 64
53 = 125
63 = 216
73 = 343
83 = 512
93 = 729

We see that our number (50) falls between 33 and 43. We shall pick the lower number, and that becomes our cube root's 10's digit.

With the 1's digit evaluating to 7 and the 10's digit evaluating to 3, it is now easy to see that the cube root of 50,653 is 37. You can check it on a calculator, if you so wish. Neat, huh?

Step 7: A Few Examples and Practice

Picture of A Few Examples and Practice

Like any trick, you will need to practice the cube root extraction technique a few times before you can really crank them out mentally. A veteran of the technique will be able to extract these numbers in ~2 seconds or less. Below are a few abbreviated examples, followed by a couple of numbers for you to try to take the cube roots of:

Example 1: 970,299
Splitting this number at the comma, we evaluate the two halves: 970, and 299. 299 ends in a 9, as does 93, so the 1's digit of the cube root is 9. 970 falls between 2 cubes: 729 (93) and 1000 (103). Picking the lower one, we get a 10's digit of 9. Thus, the cube root of 970,299 is 99.

Example 2: 91,125
Using the same split technique as above, we split the number into 2 halves: 91 and 125. 125 ends in a 5, as does 53, so the 1's digit is 5. 91 falls between 64(43) and 125(53). Picking the lower one, we get a 10's digit of 4, thus the cube root of 91,125 is 45.

Example 3: 512,000
This example adds one small caveat to our previous knowledge. As you can see, the left hand side of the cube is 512, a perfect cube. If that's the case, pick the cube that is the same as this perfect cube (in this example, 83 = 512, so we get a 10's digit of 8). Then, for the right hand side, 03 is of course 0, so the 1's digit is 0. Evaluating, we get a cube root of 80.

Here are a few more you can try, answers in the next step:
1) 2,744
2) 704,969
3) 148,877
4) 474,552
5) 24,389
6) 39,304
7) 68,921

Step 8: Answers to Step 7 Examples

1) 2,744 (Answer: 14)
2) 704,969 (Answer: 89)
3) 148,877 (Answer: 53)
4) 474,552 (Answer: 78)
5) 24,389 (Answer: 29)
6) 39,304 (Answer: 34)
7) 68,921 (Answer: 41)

Step 9: Beyond 6-Digits

Once you have mastered the basic technique, you may wish to expand your mental extractions beyond 6-digits. This can be done, but at the cost that you will have to learn additional cubes. For example, from our previous list, suppose you learned that 113 = 1331. You can apply the same technique as before, splitting on the comma between the hundreds and thousands places.

Example: 1,157,625
Doing the same technique as before, we split the number into 2 parts: 1,157 and 625. 625 ends in a 5, as does 53, so the right hand side of our cube root is 5. 1,157 falls between 1000 (103) and 1331 (113), taking the lower of these, the left hand side of the cube root is 10. So, our cube root is 105.

As you can see at this point, for every additional cube above 10 that you learn, you can derive 10 more cube roots.

I hope you have enjoyed learning this fun little trick, so for all of you math geeks out there, have fun amusing and amazing your friends.

--Purduecer

Comments

MelissaS304 (author)2017-07-01

Thanks! It's amazing! I love it!

WhatAthing (author)2015-09-25

Can't believe not a single person decided to spoil the magic... I'll be that guy then

To show this algebraically for anyone wondering:

Let "a" and "b" be numbers from 0 to 9.

Then a two digit number can be written as "10a + b"

and so its cube is (10a + b)^3 = 1000a^3 + 300ba^2 + 30ab^2 + b^3

Since all parts except b^3 is multiplied by 10, the only number affecting the last digit is b^3. So all you need to do is memorise the last digit of each cube of 0 to 9 and then compare it to the cube.

The second part we're interested in is the 1000a^3 bit. Since a^3 is multiplied by 1000 here, we only care about the 4th and above digits. Since it's clear that

(10a)^3 <= (10a + b)^3 < (10a + 10)^3,

we just need to find the two numbers "a", "a+1" such that the 4th+ digits of the cube is between those two numbers cubed, and then you have "a".

Then from there you just add 10a + b to get the number.

AhmedA31 (author)2015-01-08

Hi, Im studying a Maths degree amd this seems amazing. However, as any Mathematician would ask, why does this work? What is the proof for this in the general case? You have shown a proof in the squaring numbers page. Thank you very much.

AyushG1 (author)2014-08-14

this is EPIC at first i didnt believe wat it said but now i actually believe it.

Sankeerthi (author)2014-03-18

awesome!!!!!! Itz has become very easy to find the cube root for me. thnxxxx 4 the instructables

george1121 (author)2008-11-24

Excellent. reminds me of the story in Richard Feynmans book "Surely you are joing..."--where he beat an Abacus master by "just" doing Mental math--and said something to the effect of that one must one "know numbers". This technique will/should prove to be a motivator to kids. Keep them coming.

mahi16 (author)2008-08-11

Method for checking.
1. we get the digit sum of a no. by "adding across" the no. For instance, the digit sum 0f 13022 is 8.
2. we always reduce the digit sum to a single figure if it is not already a single fig.
3 .In "adding across" a no. we may drop out 9's. Thus if we happen to notice two digits that add up to 9, such as 2 and 9, if we ignore both of them; so the digit sum of 99019 = 1 at a glance.(If we add up 9,s we get the same result)
4. because :nine don't count" in the process, as we saw in3 step, a digit sun of 9 is the same as a digit sum of zero. The digit sum of 441,e.g = 0.

You may use this for Multiplication, Cubes , Squares etc.

you may also check weather your squared no, is correct or not.
I will explain it:-
take an example for 207.
square(207)= 42849
now add digits of LHS and RHS separately. we get
Sq(2+7) = 18
Sq(0) = 18
0=2+7
0=9
0=0

thus your calculation is correct.

take another example.
Sq(897) = 804609
sq(6) = 18
36 = 180
0=0.

WingDings (author)2008-08-10

Hmm, that sounds like fun, I'll have to practice that! This is nicely laid out. Kudos. By the way, I think you're one out by saying that a cube is "... any number that is multiplied by itself 3 times". There are only two multiplications in there - the first multiplication of a number by itself is its square, so the second multiplication by itself must be the cube, rather than the third multiplication.

pljm (author)2008-08-05

Wow, that's just too advanced for me to remember at the moment.I'm about to be going through geometry, so last year I learned how to get cube roots and such in algebra. I have a technque for squaring numbers that end in five though,(ex.35)this works easily until you get to 105 1.the last two digits will always end in 25 2.multiply the first digit to the number abouve it(in this case 3 and 4) 3. put the result in front of the 25 and you get the number.1225 is the answer you can easily see that these numbers can be reversed also, so they are easy square roots to remember

purduecer (author)pljm2008-08-06

Nice. Amusingly, I posted an Instructable on squaring numbers a few days back. Check it out if you want.

https://www.instructables.com/id/Square_2_Digit_Numbers_Mentally/

pljm (author)purduecer2008-08-06

Yeah, I saw that yesterday too. That's one I'll be able to remember

aeray (author)2008-07-24

Fascinating 'ible. As a carpenter, I often have to find square roots for stairs, rafters, etc. Do you have a trick that does not involve my fractional calculator? Thanks.

purduecer (author)aeray2008-08-04

I posted a new math trick just for you, my good man.

https://www.instructables.com/id/Square_2_Digit_Numbers_Mentally/

Hope you like it :-)

aeray (author)purduecer2008-08-04

Fascinating as well, and thanks for thinking of me, but I was looking to find square roots. Am I missing something though? Although I use and know more math than most (all) of the other carpenters I have worked with, sometimes math-wise I'm a little dense.

purduecer (author)aeray2008-08-04

Unfortunately, I do not currently have a technique for square roots. If I do happen to discover one, though, I will be sure to let you know

aeray (author)purduecer2008-08-04

Thanks. I suppose I could do some research myself, rather than whining to you about it.

mightywombat (author)aeray2008-07-25

You can pick up the book, "Secrets of Mental Math" by Arthur Benjamin and Michael Shermer. It has all sorts of tricks like this in it.

mahi16 (author)2008-07-21

Do not u think we should use VEDIC MATHEMATICS for more better results. use it, It will definitely hone my your skills

purduecer (author)mahi162008-07-21

Hmmmmm, vedic mathematics, you say? I've never heard of it, but I will certainly look into it.

mahi16 (author)purduecer2008-07-22

If you want to more about it mail to me at mkdas16@yahoo.co.in

purduecer (author)mahi162008-08-04

Hey man, got a trick on vedic math for ya, check it out!

https://www.instructables.com/id/Square_2_Digit_Numbers_Mentally/

Gonazar (author)2008-07-24

Where does this apply in real life? I mean it's an interesting and possibly useful skill in solving large cubes but when does anyone throw you such a large number and expect you to know that it's a cube and solve it?

purduecer (author)Gonazar2008-08-04

For a (potentially) more useful math skill, check out my new instructable on mentally squaring numbers. (Hope you like it!)

mightywombat (author)Gonazar2008-07-25

For the average person this probably only applies in math tests, but for an engineer, being able to do this sort of math in your head speeds up a lot of calculations. Calculus is hard enough without throwing big numbers like this in. Anything to simplify a multivariable equation!

Gonazar (author)mightywombat2008-07-25

Yeah but it only works if you know that it's a cube, "1) This trick only works for perfect cubes, it will not work for any arbitrary 6-digit number" So it's unlikely that someone will use this trick for purposes related to work where such numbers are uncommon. (100 cubes in 1,000,000 numbers)

purduecer (author)Gonazar2008-07-24

I'll admit the odds of using this skill for something other than entertainment are pretty low. However, sit tight and soon I will have a slightly more useful skill that you can possibly use.

VicArt (author)2008-07-29

Very cool. Thanks.

srilyk (author)2008-07-28

This, sir, is awesomesauce! I give it two thumbs up! (I'd give it stars if I could, but somehow I've lost that :'( )

Rodolfo (author)2008-07-27

How can I solve 856,700 ???

purduecer (author)Rodolfo2008-07-27

well...that doesn't sound like a perfect cube, and it quick check reveals it's not. So, splitting it at the comma, 856 is between 729 (93) and 1000(103) so, take the tens digit to be 9. However, not being a perfect cube, my trick incorrectly reveals 0 to be the 1's digit, which it's not. So, to solve for the above number accurately, you'll still want to use a calculator for the time being.

Arrrgh (author)2008-07-27

Nice Job!

lucianoabcd (author)2008-07-25

What a weird kind of post !!! Gives me a few ideas... Is any generalized demonstration out there? I mean, symbolically or a by-heart calculation on all possible cubes between say 0 and 999,999 ?

catwood (author)2008-07-25

Wow thats amazing! Now if I only had the willpower to memorize those 10 cubes to begin with...

JBizzle (author)2008-07-24

you are awesome! thanks for the tip. =)

MajHunter (author)2008-07-24

Very cool little trick.

idan25k (author)2008-07-24

You have earned many points on my coolness meter. (yes, math IS cool!)

lancefisher (author)2008-07-24

Nice! Now if there were a trick to tell if the number is a cube or not...

Alexis2445 (author)2008-07-24

That's wonderful! Thank you.

barnes (author)2008-07-24

Awesome! I love learning these little math tricks..and going into 3 math classes next year, I'm going to need a lot more of these tricks! Thanks!

zwild1 (author)2008-07-22

Hey right back at you! Congrats on the feature, keep up the high-quality instructables!

emmjul (author)2008-07-22

Super Instructable, well I hate school Math. But I like crazy stuff... Just brilliant

jeff-o (author)2008-07-22

Well that's pretty darned nifty. If I'd known this trick in high school, well, I'd probably be an even bigger geek. ;)

djmachet (author)2008-07-21

Brilliant! would like to try more of these techniques if you know them

Labot2001 (author)2008-07-21

This, sir, is NOT ONLY an amazing FIRST instructable, but also an amazing instructable, PERIOD. Kudos; I take my hat off to you sir! 5 stars on this one, and fav'd, too.

saites2001 (author)2008-07-21

I love it. Now if only there were an easy way to know if the number is in fact a perfect cube first.

purduecer (author)saites20012008-07-21

Yeah, unfortunately my trick cannot do that. However, you will be in the correct ballpark using the technique (for example, if the cube root is in the 50's the technique will also yield a cube root in the 50's).

zwild1 (author)2008-07-19

Just plain genius. I'm not really able to grasp the concept- crunching numbers is not my thing- but your Instructable is top-notch! Oh- And sorry I stole your nav instructable ;) but I'd love it if you have something to contribute! Up next is map reading...

purduecer (author)zwild12008-07-19

I might be up for that, give me some time to hone my nav skills and I will keep you posted :-)

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