If you have studied calculus, you undoubtedly learned the power rule to find the derivative of basic functions. However, when the function contains a square root or radical sign, such as
, the power rule seems difficult to apply. Using a simple exponent substitution, differentiating this function becomes very straightforward. You can then apply the same substitution and use the chain rule of calculus to differentiate many other functions that include radicals.




 Continuing with the square root of x function from above, the derivative can be simplified as:


 Then find the derivative of the second function:

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How do I use the chain rule?
For the equation in the article title (y = √x), you don’t need to use the chain rule, as there is not a function within a function. An example of a function that requires use of the chain rule for differentiation is y = (x^2 + 1)^7. To solve this, make u = x^2 + 1, then substitute this into the original equation so you get y = u^7. Differentiate u = x^2 + 1 with respect to x to get du/dx = 2x and differentiate y = u^7 with respect to u to get dy/du = 7u^6. Multiply dy/du by du/dx to cancel out the du and get dy/dx = 7u^6 * 2x = 14x * u^6. Substitute u = x^2 + 1 into dy/dx = 14x * u^6 to get your answer, which is dy/dx = 14x(x^2 + 1)^6.

How do I differentiate √x1 using the first principle?
Since the outer function is sqrt(x), you rewrite sqrt(x1) as (x1)^(1/2), and differentiating using the power rule gives you 1/2*(x1)^(1/21)=1/2*(x1)^(1/2)=1/(2*sqrt(x1)). You would normally use the chain rule for compositions: the derivative of the inner function, x1, is 1. 1 multiplying by anything won’t change anything, so your answer may be anything equivalent to 1/(2*sqrt(x1)).
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 About This Article
 Reader Success Stories
 Calculator, Practice Problems, and Answers
 Things You’ll Need
 About This Article
 About This Article
 What is a square root?
 How calculate a square root
 Properties and practical application of square roots
 Commonly used square roots
 Does the calculator support fractions?
 Video transcript
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About This Article
To differentiate the square root of x using the power rule, rewrite the square root as an exponent, or raise x to the power of 1/2. Find the derivative with the power rule, which says that the inverse function of x is equal to 1/2 times x to the power of a1, where a is the original exponent. In this case, a is 1/2, so a1 would equal 1/2. Simplify the result. To use the chain rule to differentiate the square root of x, read on!
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Reader Success Stories
You can multiply square roots, a type of radical expression, just as you might multiply whole numbers. Sometimes square roots have coefficients (an integer in front of the radical sign), but this only adds a step to the multiplication and does not change the process. The trickiest part of multiplying square roots is simplifying the expression to reach your final answer, but even this step is easy if you know your perfect squares.

Multiply the coefficients. A coefficient is a number in front of the radical sign. To do this, just ignore the radical sign and radicand, and multiply the two whole numbers. Place their product in front of the first radical sign.



Calculator, Practice Problems, and Answers
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We are not allowed to use a calculator, so how do I multiply a whole number by a square root?
When you multiply a whole number by a square root, you just put the two together, with the whole number in front of the square root. For example, 2 * (square root of 3) = 2(square root of 3). If the square root has a whole number in front of it, multiply the whole numbers together. So 2 * 4(square root of 3) = 8(square root of 3).

What is 2 root 3 times root 3?
√3 times √3 equals 3. Two times that is 6.

What is 4 divided by square root of 5?
(4√5)/5. Since radicals are not supposed to be in the denominator, you multiply by √5/√5 to get (4√5)/5.
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Always remember your perfect squares because it will make the process much easier!


All terms under the radicand are always positive, so you will not have to worry about sign rules when multiplying radicands.
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To multiply square roots, first multiply the radicands, or the numbers underneath the radical sign. If there are any coefficients in front of the radical sign, multiply them together as well. Finally, if the new radicand can be divided out by a perfect square, factor out this perfect square and simplify it. If you want to learn how to check your answers when you’re finished solving, keep reading the article!
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 For example, the square root of 1 is 1 because 1 multiplied by 1 equals 1 (1X1=1). However, the square root of 4 is 2 because 2 multiplied by 2 equals 4 (2X2=4). Think of the square root concept by imagining a tree. A tree grows from an acorn. Thus, it’s bigger than but related to the acorn, which was at its root. In the above example, 4 is the tree, and 2 is the acorn.
 Thus, the square root of 9 is 3 (3X3=9), of 16 is 4 (4X4=16), of 25 is 5 (5X5=25), of 36 is 6 (6X6=36), of 49 is 7 (7X7=49), or 64 is 8 (8X8=64), of 81 is 9 (9X9=81), and of 100 is 10 (10X10=100).^{[3] }

Use division to find the square root. To find the square root of a whole number, you could also divide the whole number by numbers until you get an answer that is the same as the number you used to divide the whole number.
 For example: 16 divided by 4 is 4. And 4 divided by 2 is 2, and so on. Thus, in those examples, 4 is the square root of 16, and 2 is the square root of 4.
 Perfect square roots do not have fractions or decimals because they involve whole numbers.

 N equals the number whose square root you are trying to find. It goes inside the check mark symbol.^{[5] }
 Thus, if you are trying to find the square root of 9, you should write a formula that puts the «N» (9) inside the check mark symbol (the «radical») and then present an equal sign and the 3. This means the “square root of 9 equals 3.”

Take a guess at it, and use the process of elimination. It’s tougher to figure out square roots of numbers that are not whole. But it’s possible.
 Let’s say you want to find the square root of 20. You know that 16 is a perfect square with a square root of 4 (4X4=16). Similarly, 25 has a square root of 5 (5X5=25), so the square root of 20 must fall in between 4 and 5.
 You could guess that 20’s square root is 4.5. Now, simply square 4.5 to check your guess. That means you multiply it by itself: 4.5X4.5. See if the answer is above or below 20. If the guess seems off, simply try another guess (maybe 4.6 or 4.4) and refine your guess until you hit 20.^{[6] }
 For example, 4.5X4.5 = 20.25, so logically you should try a smaller number, probably 4.4. 4.4X4.4 = 19.36. Thus, the square root of 20 must lie in between 4.5 and 4.4. How about 4.445X4.445. That’s 19.758. It’s closer. If you keep trying different numbers using this process, you will eventually get to 4.475X4.475 = 20.03. Rounding off, that’s 20.

 Then, divide your number by one of those square root numbers. Take the answer, and find the average of it and the number you divided by (average is just the sum of those two numbers divided by two). Then take the original number and divide it by the average you got. Finally, find the average of that answer with the first average you got.
 Sound complicated? It can be easiest to follow an example. For example, 10 lies in between the 2 perfect square numbers of 9 (3X3=9) and 16 (4X4=16). The square roots of those numbers are 3 and 4. So, divide 10 by the first number, 3. You will get 3.33. Now, average the 3 and 3.33 by adding them together and dividing them by 2. You will get 3.1667. Now take 10 divided by 3.1667. The answer is 3.1579. Now, average 3.1579 and 3.1667 by adding them together and dividing the sum you get by two. You will get 3.1623.
 Check your work by multiplying your answer (in this case 3.1623) by itself. Indeed, 3.1623 multiplied by 3.1623 equals 10.001.
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If I have a building that is 40 x 60 feet, how do you find out if it is square?
There are two ways to do it: (1) If you can measure the inside diagonals (from a corner to its opposite corner), the diagonals of a perfect rectangle are equal to each other; (2) Get a magnetic compass and sight it along two adjacent sides. The two directions should be exactly 90° from each other.

How do I calculate the square root without a calculator?
Use a factor tree. For example, 625 = 5 x 125 = 5 x 5 x 25 = 5 x 5 x 5 x 5. Because there are 4 fives, and we are looking for the square root, (5 x 5)(5 x 5) = 625. Therefore the square root of 625 is 25.

What is the smallest fourdigit whole number divisible by 9 that has two even and two odd digits?
The number 1089 is the answer. The way you work it out is 1008 is a 9 time table number. Then just add on nines starting at this number, and the first 9 times table number you get that has two even digits and odd digits is your answer.
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Memorizing the first few perfect squares is highly advisable:
 0^{2} = 0, 1^{2} = 1, 3^{2} = 9, 4^{2} = 16, 5^{2} = 25, 6^{2} = 36, 7^{2} = 49, 8^{2} = 64, 9^{2} = 81, 10^{2} = 100,
 Eventually learn these: 11^{2} = 121, 12^{2} = 144, 13^{2} 169, 14^{2} = 196, 15^{2} = 225, 16^{2} = 256, 17^{2} = 289…
 More easy fun: 10^{2} = 100, 20^{2} = 400, 30^{2} = 900, 40^{2} = 1600, 50^{2} = 2500, …
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To find a square root of a number without a calculator, see if you can get to that whole number by squaring smaller numbers, or multiplying a smaller number by itself. If the number is a perfect square, you will get a whole number as the square root. Otherwise, try squaring numbers with a decimal until you get as close as possible to your original number. If you want to learn how to estimate the square root of imperfect squares, keep reading the article!
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Use this online calculator to easily calculate the square root of a given number, including fractions. Quick and easy square root finder.
What is a square root?
The square root of a number answers the question «what number can I multiply by itself to get this number?». It is the reverse of the exponentiation operation with an exponent of 2, so if r^{2} = x, then we say that «r is the root of x». Finding the root of a number has a special notation called the radical symbol: √. Usually the radical spans over the entire equation for which the root is to be found. It is called a «square» root since multiplying a number by itself is called «squaring» as it is how one finds the area of a square.
For every positive number there are two square roots — one positive and one negative. For example, the square root of 4 is 2, but also 2, since 2 x 2 = 4. The negative root is always equal in value to the positive one, but opposite in sign. You can see examples in the table of common roots below. Most often when talking about «the root of» some number, people refer to the Principal Square Root which is always the positive root. This is the number our square root calculator outputs as well.
In geometrical terms, the square root function maps the area of a square onto its side length. The function √x is continuous for all nonnegative x and differentiable for all positive x.
How calculate a square root
 Start with a guess (b). If a is between two perfect squares, a good guess would be a number between those squares.
 Divide a by b: e = a / b. If the estimate e is an integer, stop. Also stop if the estimate has achieved the desired level of decimal precision.
 Get a new guess b_{1} by averaging the result of step #2 e and the initial guess b: b_{1} = (e + b) / 2
 Go to step #2 using b_{1} in place of b
For example, to find the square root of 30 with a precision of three numbers after the decimal point:
Step 1: a = 30 is between 25 and 36, which have roots of 5 and 6 respectively. Let us start with b = 5.5.
Step 2: e = a / b = 30 / 5.5 = 5.45(45). Since b is not equal to e (5.500 ≠ 5.454), continue calculation.
Step 3: b_{1} = (5.45 + 5.5) / 2 = 5.47727(27)
Step 4: e = 30 / 5.47727 = 5.477178. Since b_{1} = e = 5.477 within three position after the decimal point, stop the square rootfinding algorithm with a result of √30 = 5.47727(27).
Checking the outcome against the square root calculator output of 5.477226 reveals that the algorithm resulted in a correct solution. While the above process can be fairly tedious especially with larger roots, but will help you find the square root of any number with the desired decimal precision.
Properties and practical application of square roots
Square roots appear frequently in mathematics, geometry and physics. For example, many physical forces measured in quantities or intensities diminish inversely proportional to the square root of the distance. So, gravity between two objects 4 meters apart will be 1/√4 relative to their gravity at 0 meters. The same is true for radar energy waves, radio waves, light and magnetic radiation in general, and sound waves in gases. It is usually referred to as the «inversesquare law».
The square root is key in probability theory and statistics where it defines the fundamental concept of standard deviation.
Plotting the results from the square root function, as calculated using this square root calculator, on a graph reveals that it has the shape of half a parabola.
Commonly used square roots
Table of commonly encountered square roots:
The calculations were performed using this calculator.
Does the calculator support fractions?
Yes, simply enter the fraction as a decimal number (use dot as a separator) and you will get the corresponding root. For example, to compute the square root of 1/4 simply enter 0.25 in the number field, press «Calculate» and you will get 0.50 as ouput. If you are having difficulty converting a fraction to a decimal number, you will find our fraction to decimal converter handy.
Video transcript
I think you’re probably
reasonably familiar with the idea of a square root, but I
want to clarify some of the notation that at least, I always
found a little bit ambiguous at first. I
want to make it very clear in your head. If I write a 9 under a radical
sign, I think you know you’ll read this as the square
root of 9. But I want to make one
clarification. When you just see a number
under a radical sign like this, this means the principal
square root of 9. And when I say the principal
square root, I’m really saying the positive square root of 9. So this statement right
here is equal to 3. And I’m being clear here because
you might already know that 9 has two actual
square roots. By definition, a square root is
something— A square root of 9 is a number that, if
you square it, equals 9. 3 is a square root, but
so is negative 3. Negative 3 is also
a square root. But if you just write a radical
sign, you’re actually referring to the positive square
root, or the principal square root. If you want to refer to the
negative square root, you’d actually put a negative in front
of the radical sign. That is equal to negative 3. Or if you wanted to refer to
both the positive and the negative, both the principal and
the negative square roots, you’ll write a plus or
a minus sign in front of the radical sign. And of course, that’s equal to
plus or minus 3 right there. So with that out of the way,
what I want to talk about is the graph of the function, y
is equal to the principal square root of x. And see how it relates to the
function y is equal to x— Let me write it over here because
I’ll work on it. See how it relates to y
is equal to x squared. And then, if we have some time,
we’ll shift them around a little bit and get a better
understanding of what causes these functions to shift up
down or left and right. So let’s make a little value
table before we get out our graphing calculator. So this is for y is equal
to x squared. So we have x and y values. This is y is equal to the
square root of x. Once again, we have x and
y values right there. So let me just pick some
arbitrary x values right here, and I’ll stay in the
positive x domain. So let’s say x is equal
to 0, 1— Let me make it color coded. When x is equal to 0, what’s
y going to be equal to? Well y is x squared. 0 squared is 0. When x is 1, y is 1 squared,
which is 1. When x is 2, y is 2 squared,
which is 4. When x is 3, y is 3 squared,
which is 9. We’ve seen this before. And I could keep going. Let me add 4 here. So when x is 4, y is
4 squared, or 16. We’ve seen all of this. We’ve graphed our parabolas. This is all a bit of review. Now let’s see what happens
when y is equal to the principal square root of x. Let’s see what happens. And I’m going to pick some x
values on purpose just to make it interesting. When x is equal to 0, what’s
y going to be equal to? The principal square
root of 0? Well it’s 0. 0 squared is 0. When x is equal to 1, the
principal square root of x of 1 is just positive 1. It has another square root
that’s negative 1, but we don’t have a positive or
negative written here. We just have the principal
square root. When x is a 4, what is y? Well, the principal square
root of 4 is positive 2. When x is equal to
9, what’s y? When x is equal to 9,
the principal square root of 9 is 3. Finally, when x is equal to 16,
the principal square root of 16 is 4. So I think you already see how
these two are related. We’ve essentially just swapped
the x’s and the y’s. Well, these are the same x and
y’s, but here you have x is 2, y is 4. Here x is 4, y is 2. 3 comma 9, 9 comma 3. 4 comma 16, 16 comma 4. And that makes complete sense. If you were to square both sides
of this equation, you would get y squared is equal
to x right there. And, of course, you would want
to restrict the domain of y to positive y’s because this can
only take on positive values because this is a principal
square root. But the general idea, we just
swapped the x’s and y’s between this function and this
function right here, if you assume a domain of positive
x’s and positive y’s. Now, let’s see what the
graphs look like. And I think you might already
have a guess of— Let me just graph them here. Let me do them by hand because
I think that’s instructive sometimes before you take out
the graphing calculator. So I’m just going to stay in
the positive, in the first quadrant here. So let me graph this first. So we have the point 0, 0, the
point 1 comma 1, the point 2 comma 2, which I’m going to have
to draw it a little bit smaller than that. Let me mark this is 1, 2, 3. Actually, let me do
it like this. Let me go 1, 2, 3, 4, 5,
6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16. That’s about how far
I have to go. And then I have 1, 2, 3, 4, 5,
6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16. That’s about how far I have to
go in that direction as well. And now let’s graph it. So we have 0, 0, 1, 1,
2 comma 1, 2, 3, 4. 2 comma 4 right there. 3 comma 9. 3 comma 5, 6, 7, 8, 9. 3 comma 9 is right
about there. And then we have 4 comma 16. 4 comma 16 is going to
be right above there. So the graph of y is equal
to x squared, and we’ve seen this before. It’s going to look something
like this. We’re just graphing it in the
positive quadrant, so we get this upward opening
u just like that. Now let’s graph y is equal
to the principal square root of x. So here, once again,
we have 0, 0. We have 1 comma 1. We have 4 comma 2. 1, 2, 3, 4 comma 2. We have 9 comma 3. 5, 6, 7, 8, 9 comma 3
right about there. Then we have 16 comma 4. 16 comma 4 is right
about there. So this graph looks like that. So notice, they look like
they’re kind of flipped around the axes. This one opens along the yaxis,
this one opens along the xaxis. And once again, it makes
complete sense because we’ve swapped the x’s and the y’s. Especially if you just consider
the first quadrant. And actually, these are
symmetric around the line, y is equal to x. And we’ll talk about things like
inverses in the future that are symmetric around the
line, y is equal to x. And we can graph this better
on a regular graphing calculator. I found this on the web. I just did a quick web search. I want to give proper credit
to the people whose resource I’m using. So this is my.hrw.com/math06. You could pause this video. And hopefully, you should
be able read this. Especially if you’re looking
at it in HD. But let’s graph these
different things. Let’s graph it a little
bit cleaner than what I can do by hand. And actually, let me have some
of what I wrote there. So that should give you— OK. So let’s first just graph
y is equal to x squared. And then in green, let me
graph y is equal to the square root of x. They have some buttons here on
the right, just so you know what I’m doing. I have some buttons here on
the right: squared and the radical sign and all of that. Let me just focus on this. So let me just graph those. So first it did x squared
and then it did the square root of x. Look, if you just focus on the
first quadrant right here, you see that you get the exact same
result that I got over there, although mine
is messier. Now, just for fun and, you know,
I really didn’t do this yet with the regular
quadratics, let’s see what happens. What we need to do to shift
the different graphs. So with x squared, I’m going
to do two things. I’m going to scale the graphs
and I’m going to shift them. So that’s x squared. So let’s just focus on the x
squared and see what happens when we scale it. And then I’ll do it with the
radical sign as well. This will really work
for anything. Let’s see what happens when you
get 2 times— no, not 2 squared —2 times x squared. And let’s do another one that is
1.5 times our 0.— I could just do 0.4 actually. 0.5 times x squared. Let’s graph these right there. So x squared. So notice, our regular x
squared is just in red. If we scale it by 2, it’s still
a parabola with the vertex at the same place, but
we go up faster in both directions. And if we have 0.5 times x
squared, we still have a parabola, but we go up
a little bit slower. We have a wider opening u
because our scaling factor is lower than 1. So that’s how you kind of decide
how wide or how narrow the opening of our
parabola is. And then if you want to shift
it to the left or the right, and I want you to think
about why this is. So that’s x squared. Let’s say I want to just take
the graph of x squared and I want to shift it four
to the right. What I do is I say, x minus 4. x minus 4 squared. And if I want to shift it two
to the— Let’s say I want to shift it two to the
left. x plus 2 squared, what do we get? Notice it did exactly what I
said. x minus 4 squared was shifted four to the right. x plus 2 squared, was shifted
two to the left. And it might be unintuitive at
first, this shifting that I’m talking about. But really think about
what’s happening. Over here, the vertex is
where x is equal to 0. When you get 0 squared
up here. Now over here, the vertex
is when x is equal to 4. But when x is equal to 4,
you stick 4 in here, you get 4 minus 4. So you’re still squaring 0. 4 minus 4 is 0 and that’s
what you’re squaring. Over here, when x is equal to
negative 2— negative 2 plus 2 —you are squaring 0. So, in other words, whatever
you’re squaring, that 0 is equivalent to 4 here. Or 4 is equivalent to 0. And negative 2 is equivalent
to 0 over there. So I want you to think about
it a little bit. Another way you could think
about it, when x is equal to 1, we’re at this point
of the red parabola. But when x is equal to 5 on
the green parabola, you have 5 minus 4. Inside of the parentheses you
have a 1, just like x is equal to 1 over here, up here. So you’re at the same point
in the parabola. So I want you to think about
that a little bit. It might be a little
nonintuitive that you say minus 4 to shift to the
right and plus 2 to shift to the left. But it actually makes
a lot of sense. Now, the other interesting
thing is to shift things up and down. And that’s actually pretty
straightforward. You want to shift
this curve up. Let’s say we want to shift the
red curve up a little bit. You do x squared plus 1. Notice it got shifted up. If you want this green curve to
be shifted down by 5, put a minus 5 right there. And then you graph it and it
got shifted down by 5. If you want it to open up a
little wider than that, maybe scale it down a little bit. Scale it down and let’s
say 0.5 times that. So now the green curve will be
scaled down and it opens slower, it has a
wider opening. And the same idea can
be done with the principal square roots. So let me do that. Let me do the same idea. And the same idea actually, can
be done with any function. So let’s do the square
root of x. And in green, let’s do
the square root of x. Let’s say, minus 5. So we’re shifting it over
to the right by 5. And then let’s have the square
root of x plus 4. So we’re going to shift
it to the left by 4. Let’s shift it down by 3. And so lets graph
all of these. The square root of x. Then have the square
root of x minus 5. Notice it’s the exact same thing
as the square root of x, but I shifted it to
the right by 5. When x is equal to 5, I have
a 0 under the radical sign. Same thing as square
root of 0. So this point is equivalent
to that point. Now, when I have the square root
of x plus 4, I’ve shifted it over to the left by 4. When x is negative 4, I have
a 0 under the radical sign. So this point is equivalent
to that point. And then I subtracted 3, which
also shifted it down 3. So this is my starting point. If I want this blue square root
to open up slower, so it’ll be a little
bit narrower, I would scale it down. So here, putting a low number
will scale it down and make it more narrow because we’re
opening along the xaxis. So let me to do that. Let me make this green one—
Let me open up wider. So let me say it’s 3 times the
square root of x minus 5. So let’s graph all of these. So notice, this blue one now
opens up more narrow and this green one now opens up
a lot, I guess you could say, a lot faster. It’s scaled up. Then we could shift that one
up a little bit by 4. And then we graph it
and there you go. And notice when we graph these,
it’s not a sideways parabola because we’re
talking about the principal square root. And if you did the plus or minus
square root, it actually wouldn’t even be a valid
function because you would have two y values for
every x value. So that’s why we have to just
use the principal square root. Anyway, hopefully you found
this little talk, I guess, about the relationships with
parabolas, and/or with the x squared’s and the principal
square roots, useful. And how to shift them. And that will actually be really
useful in the future when we talk about inverses
and shifting functions.
Want to join the conversation?
There is a very nice method, by hand, that is seldom taught nowadays.
2 00 00 00 00 00
Now, if your number has several digits, you write them by pairs too (starting from the right), for example for the square root of 20451, to five digits, you would write
2 04 51 00 00 00 00 00
This is the general setting, not let’s see the method.
So you write your number, with additional zeros
2 00 00 00 00 00
You will write this one on the right, like for a division
2 00 00 00 00 00  1
Now we enter the «general step» (we will repeat this one for each pair of zeros.
You subtract $1^2$ from $2$, and write the rest ($1=21^2$) under the preceding, plus one pair of zeros:
2 00 00 00 00 00  1
1 00
On the right, there is the «current square root», we will complete it by adding digits.
Now, find the highest possible digit ($0$ to $9$) such that the operation $2d \times d$ is not greater than $100$. $2d$ denote the number obtained by concatenating twice the current square root, to the chosen digit.
For example, you try $20\times0=0$, $21\times1=21$, $22\times2=44$, $23\times3=46$, $24\times4=96$, $25\times5=125$. Stop! Too large, so the next digit is actually a $4$.
The rest is $10096=4$.
So we update the current square root on the right (chosen digit $4$), and the current rest (also $4$), and write down two more zeros:
2 00 00 00 00 00  14
1 00
4 00
This was the first step, and we will repeat exactly the same several times:
Twice the current root $14\times2=28$, and try $28d\times d$ so that it’s not greater than 400. Since $282\times2>400$, the next digit is a $1$, and we subtract $281\times1=281$ from $400$ ($400281=119$), and update both the rest and the current root (next digit $1$).
2 00 00 00 00 00  141
1 00
4 00
1 19 00
Twice the current root, so $282$, and try $282d\times d$ so that it’s not greater than $11900$. Since $2824\times4=11296$ and $2825\times5=14125$, too large, the next digit is a $4$.
The next rest is $1190011296=604$.
2 00 00 00 00 00  1414
1 00
4 00
1 19 00
6 04 00
Twice the current root, so $2828$, and we try $2828d\times d$ so that it’s not greater than $60400$. The good choice is $28282\times 2=56564$, and $6040056564=3836$ is the next rest, and next digit is $2$:
2 00 00 00 00 00  14142
1 00
4 00
1 19 00
6 04 00
38 36 00
Twice the current root is $28284$, so we try $28284d\times d$ so that it’s not greater than $383600$. The good choice is $1$: $282841\times1=282841$ is the largest possible, and $383600282841=100759$.
2 00 00 00 00 00  141421
1 00
4 00
1 19 00
6 04 00
38 36 00
10 07 59
Hence, we have the last rest, and the updated square root to $5$ digits, and we have the equation:
$$2 00 00 00 00 00=141421^2+10 07 59$$
We could of course continue, adding as many pairs of zeros as we wish. With two more pairs, you will get the next digit you are after.
In the usual setting (at least what I was taught by my father, who learned it in high school in the sixties, in France), you write the successive products under the square root like this:
2 00 00 00 00 00  141421
1 00 
4 00  24×4=96
1 19 00  281×1=281
6 04 00  2824×4=11296
38 36 00  28282×2=56564
10 07 59  282841×1=282841

That way, it’s relatively compact, yet it’s easy to do the subtractions.
Binary search for it.