How To Find The Square Root of Large Numbers Mentally

This math video tutorial explains how to find the square root of a large number mentally.

Simplifying Radical Expressions:

Multiplying Radical Expressions:

Adding & Subtracting Radicals:

Simplifying Cube Root Expressions:

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Rationalizing The Denominator:

Multiplying Radicals With Different Indices:

How To Solve Radical Equations:

Solving Equations With Cube Roots:

Solving Complex Radical Equations:

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How To Graph Radical Functions:

Domain of Radical Functions:

How To Graph Cube Root Functions:

Radical Expressions - Test Review:

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Final Exams and Video Playlists:

Full-Length Videos and Worksheets:

[lonicocopuffs]

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[minnicute]

I swear you deserve money that I give to my teacher who don't teach me at all. You're always responsible for my success. I hope you live happy life 💞😘

[networkhub]

This guy is saving my life. I don't need to spend 10 countless minutes figuring out square roots anymore. Give this man a gold medal

[Meta-Drew]

as long as you only ever run into perfect squares you'll be golden

[HurricaneEmily]

You can also add the digits of the numbers until you get one digit like if you add the digits of 13 you get 4. If you take the square of 13, it has to add to the same amount as 4x4. 4x4 is 16. 1+6 is 7. So the digits of the square of 13 also have to add to 7. The square of 13 is 169. Add the digits: 1+6+9=16, 1+6=7. So if you can’t tell by looking which is closer, add the digits of the numbers until you get one digit. Take the square and add the digits until you get one number. Look at your two choices and add their digits until you get one number and see which one matches.

Let’s say you’re deciding between 43 and 47. The number that is the square of one of these numbers is 2209. 2+2+0+9=13, 1+3=4. What’s the square root of 4? 2. So when you add the digits of the square root of 2209, it has to add to 2: 4+3=7 so it can’t be 43. 4+7=11, 1+1=2. It’s definitely 47.

The only time this doesn’t work is when you are choosing between multiples of 3. For example, 42 and 48. The digits of both their squares will add to 9. So you have to guess.

These one digit numbers that all numbers add down to are called digital roots. You can add the digits as I did or you can enter a number on a scientific calculator and hit mod 9 to get the digital root. These digits are also the remainders when you divide by 9.

Adding them and multiplying them is part of modular arithmetic.

[Monstergaming-secc]

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[RayRay-lown]

I believe this guy’s videos are pure gold...

[alishbakhan]

This guy is helping so much to cover 3 year of classes I had no idea how to solve I only knew perfect squares and now I understand finally after hours of searching the perfect video for me to understand

[Robert_H.]

The big problem with your method is that it only works if you are sure that the given root has an integer solution. If you don't know this, you should use the approximation formula Sqrt(n² + a) ≈ n + a/(2n) and (n + 0.5)² = n * (n+1) + 0.25 for better squares between the squares of natural numbers.

*RULESET*

Given: Sqrt(x)

*(1) Remove redundant powers of 10*

If x > 1000: 10 * Sqrt(x/100)

If x > 100, 000: 100 * Sqrt(x/10, 000)

If x > 10 * 10^(2k): 10^k * Sqrt(x/10^(2k))

New Radikand: y = x/10^(2k)

Advantage: 10 < y < 1000

You only need Square numbers of 1 to 31.

*(2) Find the closest square number n*

Bonus: If the number is relatively centered between two square numbers n1 and n2, you can form the product of these and add 0.25 (n1 * n2 + 0.25) to use the square number (n1 + 0.5)².

*(3) Calculate the difference*

a = y - n²

*(4) Use approximation formula*

Sqrt(y) = Sqrt(n² + a) ≈ n + a/(2n)




Examples:

Sqrt(1156) = Sqrt(11.56 * 100) = 10 * Sqrt(11.56) = 10 * Sqrt(12.25 - 0.69) = 10 * Sqrt(3.5² - 0.69) ≈ 10 * (3.5 - 0.69/(2*3.5)) ≈ 10 * 3.4 = 34

Sqrt(2304) = Sqrt(23.04 * 100) = 10 * Sqrt(23.04) = 10 * Sqrt(25 - 1.96) = 10 * Sqrt(5² - 1.96) ≈ 10 * (5 - 1.96/(2*5)) = 10 * 4.8 = 48

Sqrt(4489) = Sqrt(44.89 * 100) = 10 * Sqrt(44.89) = 10 * Sqrt(42.25 + 2.64) = 10 * Sqrt(6.5² + 2.64) ≈ 10 * (6.5 + 2.64/(2*6.5)) ≈ 10 * 6.7 = 67

Sqrt(12996) = Sqrt(129.96 * 100) = 10 * Sqrt(129.96) = 10 * Sqrt(132.25 - 2.29) = 10 * Sqrt(11.5² - 2.29) ≈ 10 * (11.5 - 2.29/(2*11.5)) ≈ 10 * 11.4 = 114

Sqrt(24649) = Sqrt(246.49 * 100) = 10 * Sqrt(246.49) = 10 * Sqrt(240.25 + 6.24) = 10 * Sqrt(15.5² + 6.24) ≈ 10 * (15.5 + 6.24/(2*15.5)) ≈ 10 * 15.7 = 157

And for error estimation for the approximation formula you can use the following term:

Error = a²/(8n³)

With a < n:

Error < n²/(8n³) = 1/(8n)



Without the 0.5 formula:

Sqrt(1156) = Sqrt(11.56 * 100) = 10 * Sqrt(11.56) = 10 * Sqrt(9 + 2.56) = 10 * (3² + 2.56) ≈ 10 * (3 + 2.56/(2*3)) ≈ 10 * 3.4 = 34

Sqrt(4489) = Sqrt(44.89 * 100) = 10 * Sqrt(44.89) = 10 * Sqrt(49 - 4.11) = 10 * Sqrt(7² - 4.11) ≈ 10 * (7 - 4.11/(2*7)) ≈ 10 * 6.7 = 67

Sqrt(12996) = Sqrt(129.96 * 100) = 10 * Sqrt(129.96) = 10 * Sqrt(121 + 8.96) = 10 * Sqrt(11² + 8.96) ≈ 10 * (11 + 8.96/(2*11)) ≈ 10 * 11.4 = 114

Sqrt(24649) = Sqrt(246.49 * 100) = 10 * Sqrt(246.49) = 10 * Sqrt(256 - 9.51) = 10 * Sqrt(16² - 9.51) ≈ 10 * (16 - 9.51/(2*16)) ≈ 10 * 15.7 = 157

[decentassassin]

Hey man, I'm speaking as young as 13! Amazing strat, I honestly would ever be amazed to have you as my math teacher but I appreciate the fact that I took this lesson online. I look forward to more of your videos, please keep it up! :D

[lucasmacdonald]

Dude you couldn’t have posted this at a better time

[Overrunnerr]

How did you get square root 9 and 16 from square root 11? This part confused me a lot.

For anyone who is confused, Square root 11 is between perfect square root of 9 and 16 . If it was square root of 30, it would be between perfect squares of 25 and 36.

All perfect squares are:
1
4
9
16
25
36
49
64
81
etc...

[premamuthukumar]

Lovely teaching! Also instead of doing: For example on the square root of 2304:

its either 42 or 48 and instead of finding the square root of 40 and 50, you can find the square root of 45 which is in the middle of both numbers.

An easy way to work out how to square numbers ending in 5:
5 x 5 is *25, * then keep that.
for 45, you have done the 5 and you have 4 left.
Add one to that number then times both numbers: 4 times 5 which gives you *20.*

Put the numbers together in the order they were in:
4 5
20 25
Your answer is 2025

The original number was 2304 (The number that you had to work out the square root of) which is higher than 2025 so the answer has to be higher too. The answer is 48.

Sorry If I didnt explain well, I just think this is a helpful way to do it. It is a bit faster. :)

[jojoiii]

in grade 12 and regret never practicing these properly and was searching a proper and a clear way to do these. thank you so much. helpful as always!

[douglasmagowan]

One more trick, that really should be here... Taking the example of the square root of 4489, rather than asking if this is closer to 60^2 or 70^2, we consider 65^2 = 60*70 + 25

[fusion]

this is my favorite channel. I love this stuff

[maydad]

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[mattslupek]

There’s another way to do it, too. After determining the two possible digits in the ones column, ignore the last two digits, and find the nearest square below the remaining numbers and find the root. That’s the number for the tens column.

Multiply that number by the next number up. If the product is smaller than the number left of the ignored digits, pick the larger number for the ones column.

[aianna]

1:50 i found something wrong take a look at the 15² it says 215 but the correct one is actually 225

[KJAAICH]

Ur voice says that ur mind is so clear about this complicated presentation thank you