Square Root Calculator
- Square root symbol √
- Square root definition
- How to find the square root?
- Square root calculator
- How to simplify square roots?
- Adding, subtracting, multiplying and dividing square roots
- Square roots of exponents and fractions
- Square root function and graph
- Derivative of the square root
- Square root of a negative number
- FAQ
Our square root calculator estimates the square root of any positive number you want. Just enter the chosen number and read the results. Everything is calculated quickly and automatically! With this tool, you can also estimate the square of the desired number (just enter the value into the second field) which may be a great help in finding perfect squares from the square root formula.
Are you struggling with the basic arithmetic operations: adding square roots, subtracting square roots, multiplying square roots or dividing square roots? Not any more! In the following text, you will find a detailed explanation about different square root properties, e.g., how to simplify square roots, with many various examples given. With this article, you will learn once and for all how to find square roots!
Have you ever wondered what is the origin of the square root symbol √? We can assure you that this history is not as simple as you might think at first. The origin of the root symbol goes back to ancient times, as the origin of the percent sign.
If you're looking for the square root graph or square root function properties, head directly to the appropriate section (just click the links above!). There, we explain what is the derivative of a square root using a fundamental square root definition; we also elaborate on how to calculate square roots of exponents or square roots of fractions. Finally, if you are persistent enough, you will find out that square root of a negative number is, in fact, possible. In that way, we introduce complex numbers which find broad applications in physics and mathematics.
Square root symbol √
The operation of the square root of a number was already known in antiquity. The earliest clay tablet with the correct value of up to 5 decimal places of √2 = 1.41421 comes from Babylonia (1800 BC - 1600 BC). Many other documents show that square roots were also used by the ancient Egyptians, Indians, Greeks, and Chinese. However, the origin of the root symbol √ is still largely speculative.
- many scholars believe that square roots originate from the letter "r" - the first letter of the Latin word radix meaning root,
- another theory states that square root symbol was taken from the Arabic letter ج that was placed in its original form of ﺟ in the word جذر - root (the Arabic language is written from right to left).
The first use of the square root symbol √ didn't include the horizontal "bar" over the numbers inside the square root (or radical) symbol, √‾. The "bar" is known as a vinculum in Latin, meaning bond. Although the radical symbol with vinculum is now in everyday use, we usually omit this overline in the many texts, like in articles on the internet. The notation of the higher degrees of a root has been suggested by Albert Girard who placed the degree index within the opening of the radical sign, e.g., ³√ or ⁴√.
The last question is why is the square root operation called root regardless of its true origin? The explanation should become more evident if we write the equation x = ⁿ√a in a different form: xⁿ = a. x is called a root or radical because it is the hidden base of a. Thus, the word radical doesn't mean far-reaching or extreme, but instead foundational, reaching the root cause.
Square root definition
In mathematics, the traditional operations on numbers are addition, subtraction, multiplication, and division. Nonetheless, we sometimes add to this list some more advanced operations and manipulations: square roots, exponentiation, logarithmic functions and even trigonometric functions (e.g., sine and cosine). In this article, we will focus on the square root definition only.
The square root of a given number x
is every number y
whose square y² = y*y
yields the original number x
. Therefore, the square root formula can be expressed as:
√x = y ⟺ x = y²
,
where ⟺
is a mathematical symbol that means if and only if. Each positive real number always has two square roots - the first is positive and second is negative. However, for many practical purposes, we usually use the positive one. The only number that has one square root is zero. It is because √0 = 0 and zero is neither positive nor negative.
There is also another common notation of square roots that could be more convenient in many complex calculations. This alternative square root formula states that the square root of a number is a number raised to the exponent of the fraction one half:
√x = x^(1/2) = x^(0.5)
🔎 Is 1/2 a fraction or a ratio? Or maybe both? Learn the difference with our ratio calculator!
In geometric interpretation, the square root of a given area of a square gives the length of its side. That's why √
has word square in its name. A similar situation is with the cube root ∛
. If you take the cube root of the volume of a cube, you get the length of its edges. While square roots are used when considering surface areas, cube roots are useful to determine quantities that relate to the volume, e.g., density.
How to find the square root?
Maybe we aren't being very modest, but we think that the best answer to the question how to find the square root is straightforward: use the square root calculator! You can use it both on your computer and your smartphone to quickly estimate the square root of a given number. Unfortunately, there are sometimes situations when you can rely only on yourself, what then? To prepare for this, you should remember several basic perfect square roots:
- square root of 1:
√1 = 1
, since1 * 1 = 1
; - square root of 4:
√4 = 2
, since2 * 2 = 4
; - square root of 9:
√9 = 3
, since3 * 3 = 9
; - square root of 16:
√16 = 4
, since4 * 4 = 16
; - square root of 25:
√25 = 5
, since5 * 5 = 25
; - square root of 36:
√36 = 6
, since6 * 6 = 36
; - square root of 49:
√49 = 7
, since7 * 7 = 49
; - square root of 64:
√64 = 8
, since8 * 8 = 64
; - square root of 81:
√81 = 9
, since9 * 9 = 81
; - square root of 100:
√100 = 10
, since10 * 10 = 100
; - square root of 121:
√121 = 11
, since11 * 11 = 121
; - square root of 144:
√144 = 12
, since12 * 12 = 144
;
The above numbers are the simplest square roots because every time you obtain an integer. Try to remember them! But what can you do when there is a number that doesn't have such a nice square root? There are multiple solutions. First of all, you can try to predict the result by trial and error. Let's say that you want to estimate the square root of 52
:
- You know that
√49 = 7
and√64 = 8
so√52
should be between7
and8
. - Number
52
is closer to the49
(effectively closer to the7
) so you can try guessing that√52
is7.3
. - Then, you square
7.3
obtaining7.3² = 53.29
(as the square root formula says) which is higher than52
. You have to try with a smaller number, let's say7.2
. - The square of
7.2
is51.84
. Now you have a smaller number, but much closer to the52
. If that accuracy satisfies you, you can end estimations here. Otherwise, you can repeat the procedure with a number chosen between7.2
and7.3
,e.g.,7.22
and so on and so forth.
Another approach is to simplify the square root first and then use the approximations of the prime numbers square roots (typically rounded to two decimal places):
- square root of 2:
√2 ≈ 1.41
, - square root of 3:
√3 ≈ 1.73
, - square root of 5:
√5 ≈ 2.24
, - square root of 7:
√7 ≈ 2.65
, - square root of 11:
√11 ≈ 3.32
, - square root of 13:
√13 ≈ 3.61
, - square root of 17:
√17 ≈ 4.12
, - square root of 19:
√19 ≈ 4.34
, etc.
Let's try and find the square root of 52
again. You can simplify it to √52 = 2√13
(you will learn how to simplify square root in the next section) and then substitute √13 ≈ 3.61
. Finally, make a multiplication √52 ≈ 2 * 3.61 = 7.22
. The result is the same as before!
You can check whether a number is prime or not with our prime number calculator. A prime number is a natural number (greater than one) that can't be obtained as a product of two smaller natural numbers. For example, 7 is a prime number because you can get it only by multiplying 1 * 7
or 7 * 1
. On the other hand, number 8 is not prime, because you can form it by multiplying 2 * 4
or 4 * 2
(besides product of 1 and 8 itself).
Square root calculator
In some situations, you don't need to know the exact result of the square root. If this is the case, our square root calculator is the best option to estimate the value of every square root you desired. For example, let's say you want to know whether 4√5
is greater than 9
. From the calculator, you know that √5 ≈ 2.23607
, so 4√5 ≈ 4 * 2.23607 = 8.94428
. It is very close to the 9
, but it isn't greater than it! The square root calculator gives the final value with relatively high accuracy (to five digits in above example). With the significant figure calculator, you can calculate this result to as many significant figures as you want.
Remember that our calculator automatically recalculates numbers entered into either of the fields. You can find what is the square root of a specific number by filling the first window or get the square of a number that you entered in the second window. The second option is handy in finding perfect squares that are essential in many aspects of math and science. For example, if you enter 17
in the second field, you will find out that 289
is a perfect square.
In some applications of the square root, particularly those pertaining to sciences such as chemistry and physics, the results are preferred in scientific notation. In brief, an answer in scientific notation must have a decimal point between the first two non-zero numbers and will be represented as the decimal multiplied by 10 raised to an exponent. For example, the number 0.00345
is written as 3.45 * 10⁻³
in scientific notation, whereas 145.67
is written as 1.4567 * 10²
in scientific notation. The results obtained using the square root calculator can be converted to scientific notation with the scientific notation calculator.
How to simplify square roots?
First, let's ask ourselves which square roots can be simplified. To answer it, you need to take the number which is after the square root symbol and find its factors. If any of its factors are square numbers (4, 9, 16, 25, 36, 49, 64 and so on), then you can simplify the square root. Why are these numbers square? They can be respectively expressed as 2², 3², 4², 5², 6², 7² and so on. According to the square root definition, you can call them perfect squares. We've got a special tool called the factor calculator which might be very handy here. Let's take a look at some examples:
- can you simplify √27? With the calculator mentioned above, you obtain factors of 27: 1, 3, 9, 27. There is 9 here! This means you can simplify √27.
- can you simplify √15? Factors of 15 are 1, 3, 5, 15. There are no perfect squares in those numbers, so this square root can't be simplified.
So, how to simplify square roots? To explain that, we will use a handy square root property we have talked about earlier, namely, the alternative square root formula:
√x = x^(1/2)
We can use those two forms of square roots and switch between them whenever we want. Particularly, we remember that power of multiplication of two specific numbers is equivalent to the multiplication of those specific numbers raised to the same powers. Therefore, we can write:
(x * y)^(1/2) = x^(1/2) * y^(1/2) ⟺ √(x * y) = √x * √y
,
How can you use this knowledge? The argument of a square root is usually not a perfect square you can easily calculate, but it may contain a perfect square amongst its factors. In other words, you can write it as a multiplication of two numbers, where one of the numbers is the perfect square, e.g., 45 = 9 * 5
(9 is a perfect square). The requirement of having at least one factor that is a perfect square is necessary to simplify the square root. At this point, you should probably know what the next step will be. You need to put this multiplication under the square root. In our example:
√45 = 45^(1/2) = (9 * 5)^(1/2) = 9^(1/2) * 5^(1/2) = √9 * √5 = 3√5
.
You have successfully simplified your first square root! Of course, you don't have to write down all these calculations. As long as you remember that square root is equivalent to the power of one half, you can shorten them. Let's practice simplifying square roots with some other examples:
- How to simplify square root of 27?
√27 = √(9 * 3) = √9 * √3 = 3√3
; - How to simplify square root of 8?
√8 = √(4 * 2) = √4 * √2 = 2√2
; - How to simplify square root of 144?
√144 = √(4 * 36) = √4 * √36 = 2 * 6 = 12
.
In the last example, you didn't have to simplify the square root at all, because 144 is a perfect square. You could just remember that 12 * 12 = 144. However, we wanted to show you that with the process of simplification, you can easily calculate square roots of perfect squares too. It is useful when dealing with big numbers.
Finally, you may ask how to simplify roots of higher orders, e.g., cube roots. In fact, the process is very analogical to the square roots, but in the case of cube roots, you have to find at least one factor that is a perfect cube, not a perfect square, i.e., 8 = 2³, 27 = 3³, 64 = 4³, 125 = 5³ and so on. Then you divide your number into two parts and put under the cube root. Let's take the following example of simplifying ³√192:
∛192 = ∛(64 * 3) = ∛64 * ∛3 = 4∛3
It may seem a little bit complicated at first glance, but after some practice, you will be able to simplify roots in your head. Trust us!
Adding, subtracting, multiplying and dividing square roots
Adding square roots and subtracting square roots
Unfortunately, adding or subtracting square roots are not as easy as adding/subtracting regular numbers. For example, if 2 + 3 = 5, it doesn't mean that √2 + √3 equals √5. That's wrong! To understand why is that, imagine that you have two different types of shapes: triangles 🔺 and circles 🔵. What happens when you add one triangle to one circle 🔺 + 🔵? Nothing! You still have one triangle and one circle 🔺 + 🔵. On the other hand, what happens when you try to add three triangles to five triangles: 3🔺 + 5🔺? You'll we get eight triangles 8🔺.
Adding square roots is very similar to this. The result of adding √2 + √3 is still √2 + √3. You can't simplify it further. It is a different situation however when both square roots have the same number under the root symbol. Then we can add them just as regular numbers (or triangles). For example 3√2 + 5√2 equals 8√2. The same thing is true subtracting square roots. Let's take a look at more examples illustrating this square root property:
- What is
6√17 + 5√17
? Answer:6√17 + 5√17 = 11√17
; - What is
4√7 - 7√7
? Answer:4√7 - 7√7 = -3√7
; - What is
2√2 + 3√8
? Answer:2√2 + 3√8 = 2√2 + 6√2 = 8√2
, because we simplified √8 = √(4 * 2) = √4 * √2 = 2√2; - What is
√45 - √20
? Answer:√45 - √20 = 3√5 - 2√5 = √5
, because we simplified √45 = √(9 * 5) = √9 * √5 = 3√5 and √20 = √(4 * 5) = √4 * √5 = 2√5; - What is
7√13 + 2√22
? Answer:7√13 + 2√22
, we can't simplify this further; - What is
√3 - √18
? Answer:√3 - √18 = √3 - 3√2
, we can't simplify this further than this, but we at least simplified √18 = √(9 * 2) = √9 * √2 = 3√2.
Multiplying square roots and dividing square roots
Now, when adding square roots is a piece of cake for you, let's go one step further. What about multiplying square roots and dividing square roots? Don't be scared! In fact, you already did it during the lesson of simplifying square roots. Multiplying square roots is based on the square root property that we have used before a few times, that is:
√x = x^(1/2)
Do you remember how to multiply numbers that are raised to the same power? As a reminder:
xⁿ * yⁿ = (x * y)ⁿ
,
and therefore
x^(1/2) * y^(1/2) = (x * y)^(1/2) ⟺ √x * √y = √(x * y)
.
As opposed to addition, you can multiply every two square roots. Remember that multiplication has commutative properties, that means that the order to which two numbers are multiplied does not matter. Few examples should clarify this issue:
- What is
√3 * √2
? Answer:√3 * √2 = √6
; - What is
2√5 * 5√3
? Answer:2√5 * 5√3 = 2 * 5 * √5 * √3 = 10√15
, because multiplication is commutative; - What is
2√6 * 3√3
? Answer:2√6 * 3√3 = 2 * 3 * √6 * √3 = 6√18 = 18√3
, we simplified √18 = √(9 * 2) = √9 * √2 = 3√2.
Dividing square root is almost the same since:
x^(1/2) / y^(1/2) = (x / y)^(1/2) ⟺ √x / √y = √(x / y)
.
All you need to do is to replace multiplication sign with a division. However, the division is not a commutative operator! You have to calculate the numbers that stand before the square roots and numbers under the square roots separately. As always, some practical examples:
- What is
√15 / √3
? Answer:√15 / √3 = √5
; - What is
10√6 / 5√2
? Answer:10√6 / 5√2 = (10 / 5) * (√6 / √2) = 2√3
; - What is
6√2 / 3√5
? Answer:6√2 / 3√5 = (6 / 3) * (√2 / √5) = 2√(2/5) = 2√(0.4)
, we switched there from a simple fraction 2/5 to the decimal fraction 2/5 = 4/10 = 0.4.
Square roots of exponents and fractions
Calculating the square root of the exponent or square root of the fraction might not be clear for you. But with the knowledge you acquired in the previous section, you should find it easier than you expected! Let's begin with the square roots of exponents. In that case, it will be easier for you to use the alternative form of square root √x = x^(1/2)
. Do you remember the power rule? If not, here is a quick reminder:
(x^n)^m = x^(n*m)
,
where n
and m
are any real numbers. Now, when you place 1/2
instead of m
you'll get nothing else but a square root:
√(x^n) = (x^n)^(1/2) = x^(n/2)
,
and that's how you find the square root of an exponent. Speaking of exponents, above equation looks very similar to the standard normal distribution density function, which is widely used in statistics.
If you're still not sure about taking square roots of exponents, here are a few examples:
- square root of 2^4:
√(2^4) = (2^4)^(1/2) = 2^(4/2) = 2^2 = 4
, - square root of 5^3:
√(5^3) = (5^3)^(1/2) = 5^(3/2)
, - square root of 4^5:
√(4^5) = (4^5)^(1/2) = 4^(5/2) = (2^2)^(5/2) = 2^5 = 32
.
As you can see, sometimes it is impossible to get a pretty result like the first example. However, in the third example, we showed you a little trick with expressing 4
as 2^2
. This approach can often simplify more complicated equations.
What about square roots of fractions? Take a look at the previous section where we wrote about dividing square roots. You can find there the following relation that should explain everything:
(x / y)^(1/2) ⟺ √x / √y = √(x / y)
,
where x / y
is a fraction. Below you can find some examples of square roots of a fraction:
- square root of 4/9:
√(4/9) = √4 / √9 = 2/3
, - square root of 1/100:
√(1/100) = √1 / √100 = 1/10
, - square root of 1/5:
√(1/5) = √1 / √5 = 1/√5 = √5/5
.
Leaving roots in the denominator is not a very good habit. That's why we got rid of it in the last example. We just multiplied both the numerator and denominator by the same number (we can always do that, as the number we multiply by equals 1), in this case by √5
.
Square root function and graph
Functions play a vital role not only in mathematics but in many other areas like physics, statistics, or finance. Function f(x)
is nothing more than a formula that says how the value of f(x)
changes with the argument x
. To see some examples, check out our finance tools made by financial specialists, for example, the compound interest calculator or future value calculator. You will find there some functions that you can apply in real life. They're a great help if you want to know how to calculate the compound interest or to estimate the future value of an annuity.
Below you can find the square root graph, made up of half of a parabola. Check it and try to validate, for example, whether the square root function of x = 9
is 3
and of x = 16
is 4
(as it should be).
Let's go back to the square root function f(x) = √x
and explore what are its basic properties. We consider there only the positive part of f(x)
(as you can see in the square root graph above). So, the square root function:
- is continuous and growing for all non-negative
x
, - is differentiable for all positive
x
(see the section for more information), - approaches the limit of infinity as
x
approaches infinity (lim √x → ∞
whenx → ∞
), - is a real number for all non-negative
x
and a complex number for all negativex
(we write more about it in the section).
You probably have already noticed that the square root of the area of a square gives its side length. This feature is used in one of our construction calculators - square footage calculator. If you plan to do any renovation in the future, these tools might be a great help. Don't forget to use them!
Derivative of the square root
A derivative of a function tells us how fast this function changes with its argument. One of the simplest examples in velocity (the rate of change of position). Let's say that the function x(t)
describes how the distance of the moving car from a specific point changes with time t
. Do you know what determines how fast the change is in your distance traveled? The answer is the speed of the car! So the derivative of the position x(t)
is velocity v(t)
(velocity can depend on time too). To denote derivative, we usually use apostrophe v(t) = x'(t)
or the derivative symbol v(t) = dx(t)/dt
.
The derivative of the general function f(x)
is not always easy to calculate. However, in some circumstances, if the function takes a specific form, we've got some formulas. For example, if
f(x) = x^n
,
where n
is any real number, the derivative is as follows:
f'(x) = n * x^(n-1)
.
It may not look like, but this answers the question what is the derivative of a square root. Do you remember the alternative (exponential) form of a square root? Let us remind you:
√x = x^(1/2)
.
You can see that in this case n = 1/2
, so the derivative of a square root is:
(√x)' = (x^(1/2))' = 1/2 * x^(-1/2) = 1/(2√x)
.
Since a number to a negative power is one over that number, the estimation of the derivation will involve fractions. We've got a tool that could be essential when adding or subtracting fractions with different denominators. It is called the LCM calculator, and it tells you how to find the Least Common Multiple.
The derivative of a square root is needed to obtain the coefficients in the so-called Taylor expansion. We don't want to dive into details too deeply, so, briefly, the Taylor series allows you to approximate various functions with the polynomials that are much easier to calculate. For example, the Taylor expansion of √(1 + x)
about the point x = 0
is given by:
√(1 + x) = 1 + 1/2 * x - 1/8 * x² + 1/16 * x³ - 5/128 * x⁴ + ...
,
which is valid for -1 ≤ x ≤ 1
. Although the above expression has an infinite number of terms, to get the approximate value you can use just a few first terms. Let's try it! With x = 0.5
and first five terms, you get:
√(1.5) = 1 + 1/2 * 0.5 - 1/8 * 0.25 + 1/16 * 0.125 - 5/128 * 0.0625
,
√(1.5) ≈ 1.2241
,
and the real value, provided by our calculator, is √(1.5) ≈ 1.2247
. Close enough!
That was a lot of maths and equations so far. For those of you who are persistent enough, we've prepared the next section which explains how to calculate the square root of a negative number.
Square root of a negative number
At school, you probably have been taught that square root of a negative number does not exist. This is true when you consider only real numbers. A long time ago, to perform advanced calculations, mathematicians had to introduce a more general set of numbers - the complex numbers. They can be expressed in the following form:
x = a + b*i
,
where x
is the complex number with the real part a
and imaginary part b
. What differs between a complex number and a real one is the imaginary number i
. Here you have some examples of complex numbers: 2 + 3i
, 5i
, 1.5 + 4i
, 2
. You may be surprised seeing 2
there which is a real number. Yes, it is, but it is also a complex number with b = 0
. Complex numbers are a generalization of the real numbers.
So far imaginary number i
is probably still a mystery for you. What is it at all? Well, although it may look weird, it is defined by the following equation:
i = √(-1)
,
and that's all that you need to calculate the square root of every number, whether it is positive or not. Let's see some examples:
- square root of -9:
√(-9) = √(-1 * 9) = √(-1)√9 = 3i
, - square root of -13:
√(-13) = √(-1 * 13) = √(-1)√13 = i√13
, - square root of -49:
√(-49) = √(-1 * 49) = √(-1)√49 = 7i
.
Isn't that simple? This problem doesn't arise with the cube root since you can obtain the negative number by multiplying three of the identical negative numbers (which you can't do with two negative numbers). For example:
³√(-64) = ³√[(-4)*(-4)*(-4)] = -4
.
That's probably everything you should know about square roots. We appreciate that you stayed with us until this point! As a reward you should bake something sweet for yourself :-) Check out our perfect pancake calculator to find out how to make the perfect pancake, however you like it. You may need our grams to cups calculator to help you with this. It works both ways, i.e., to convert grams to cups and convert cups to grams. And if you ask yourself "How many calories should I eat a day?", visit our handy calorie calculator!
FAQ
Can a number have more than one square root?
Yes, in fact all positive numbers have 2 square roots, one that is positive and another that is equal but negative to the first. This is because if you multiply two negatives together, the negatives cancel and the result is positive.
How do you find the square root without a calculator?
- Make an estimate of the square root. The closest square number is acceptable if you’re at a loss.
- Divide the number you want to find the square root of by the estimate.
- Add the estimate to the result of step 2.
- Divide the result of step 3 by 2. This is your new estimate.
- Repeat steps 2-4 with you new estimate. The more times this is repeated, the more accurate the result is.
How can I estimate square roots?
- Find the nearest square number above and below the number you are thinking of.
- The square root will be between the square roots of these numbers.
- The closeness of the number to a square root indicates how close the root is. E.g., 26 is very close to 25, so the root will be very close to 5.
- Try a few times to get the hang of it.
Is the square root of 2 a rational number?
No, the square root of 2 is not rational. This is because when 2 is written as a fraction, ^{2}/_{1}, it can never have only even exponents, and therefore a rational number cannot have been squared to create it.
How can I get rid of a square root?
In algebra, squaring both sides of the equation will get rid of any square roots. The result of this operation is that the square roots will be replaced with whatever number they were finding the square root of.
Are square roots rational?
Some square roots are rational, whereas others are not. You can work out if a square root is rational or not by finding out if the number you are square rooting can be expressed in terms of only even exponents (e.g. 4 = 2^{2} / 1 ^{2}). If it can, its root is rational.
Is the square root of 5 a rational number?
The square root of 5 is not a rational number. This is because 5 cannot be expressed as a fraction where both the numerator and denominator have even exponents. This means that a rational number cannot have been squared to get 5.
Is the square root of 7 a rational number?
The result of square rooting 7 is an irrational number. 7 cannot be written as a fraction with only even exponents, meaning that the number squared to reach 7 cannot be expressed as a fraction of integers, and therefore is not rational.
What is the derivative of the square root of x?
The derivative of square root x is ^{x-1/2}/_{2}, or ^{1}/_{2SQRT(x)}. This is because the square root of x can be expressed as x^{1/2}, from which differentiation occurs normally.
How do you find the square root of a decimal?
- Convert the decimal into a fraction.
- Find any square roots of the fraction, or estimate it. Make the fraction equal to the square root you found squared.
- Cancel the square root and the square leaving you with the fraction.
- Rewrite the fraction as a decimal as your final answer.