What is the **square root of 3** or **√3**? It is strange, isn’t it! It is totally different from the square root of 16 or 25 or 36 and so on. We can easily calculate the square root of 16 or 25 or 36 but what about if we ask for the square root of 3. Is it possible at all? Yes, it is possible! Let’s discuss, what is the *square root of 3* with simple steps!

## What is the Square Root of 3?

3^{1/2} or 3^{0.5}, respectively, is the radical form of the square root of 3. Rounding up to 7 decimal places, 3’s square root is 1.7320508. Thus, x^{2} = 3 is a positive equation.

- The square Root of 3 is 1.7320508075688772.
- The square Root of 3 in exponential form is 3
^{½}or 3^{0.5}. - The square Root of 3 in radical form is √3.

Let us try to learn, what is the square root of 3, it’s a numerical value. If we try to calculate √3 in a calculator, what will we see? A very long list of irrational numbers comes after the decimal point and we cannot remember it. So, what do we do normally in our calculations? Simply write √3.

Square root refers to the number that is produced when multiplying itself by itself. Five times five results in 25, which is the square root of 25. Nevertheless, some numbers may yield square roots that do not result in whole numbers, for example, 3. There are several ways to express the square root of 3. Many times, writing √3, cannot help us to get the numerical value and we must find out the numerical value of the square root of 3.

## Square Root of 3 Basics

The square root of 3 means a number when we multiply with the same number, the result comes of 3. We can write, √**Y** x √**Y** =**Y** It is written by √3. So, as per definition, we can say, √3x√3 =3

There are thousands of research by mathematicians for the digits after decimal points in the square root of 3. In many types of research, the result shows even a billion digits after decimals. Let’s see, the numerical value of √3 up to 15 decimal places, **√3 = 1.732050807568880**

## History of Square Root of 3

Various methods of determining the square root of numbers were invented by Babylonians in the early 1900s as a result of the invention of squares and square roots. The number below gives ‘What is the value of root 3’ determined up to 4 decimal places. Indian Mathematicians have recorded a number of ways to find the square root of numbers in their works, such as the Sulabha Sutra by Baudhayana, and the Aryabhatiya by Aryabhatta.

## Example of the Square Root of 3

### Square Root of 3 Examples

We use the square root of 3 or √3 in many ways. Let’s see a few examples of the use of √3. Let us consider an equation to solve the value of y,

- 7x
^{2}=3x^{2}+12 - Or, 4x
^{2}=12 - Or, x
^{2 }=12/4=3 - Or, x = √3

We used to write,

- 2+√3
- 6+√3
- 2-√3

Or sometimes, we multiply,

- 5x√3=5√3
- √3x√3x√3=3√3
- 10x6x√3=60√3 and so on.

Hence, it is understood that whenever we got the square root of 3, we simply write the symbol √3. Now, how can we specify the numerical value of √3? For example, if someone asks you to bring 5√3kg of rice, will you be able to bring that quantity? How do you calculate? Yes! It can be possible if you are able to find out the numerical value of the square root of 3. Few More Examples of the use of the square root of 3:

### What is the square root of 3 divided by 3?

The square root of 3 divided by 3 means, Square root of 3/3 or √3/3

Now, what is the value of √3/3 = ??

We know that 3 = √3 x √3

Hence, the value becomes,

- Value = √3/3 = √3 / (√3 x √3) [ as 3 = √3 x √3]
- Value = 1/√3

We have seen that the value square root of 3 is 1.732

Hence, value = 1/√3 = 1/1.732 = 0.577

## How to Find the Square Root of 3?

There are a lot of methods to find out the square root of 3. Few easy methods are as follows,

- Long division method
- Average method
- Equation method

**Square root of 5** also can be calculated.

### Long Division Method for Square Root 3

The long division method is the best method to find out the square root of any number other than a perfect square. It’s a little lengthy but very easy to calculate and due to the lengthy steps, it is known as the long division method. Let’s explore all the steps along with an explanation,

**Step#1: Write the number 3 as 3.00000000. Here, we have just changed the format only, values are the same for both the numbers. **

**Step#2: Zeros need to be kept in order like, 3.00 00 00 00. Here also, value is same and format is little changed so that it will be easy to understand the long division method.**

**Step#3: Now, the third step is to find out the perfect square just below 3. If we analyze, 0,1,2, are numbers below 3, and 1 is the perfect square.** As we know, 1 = 1×1 = 1^{2}

**Step#4: We need to find out the square root of the perfect square, below 3. Here, it is 1 and the square root of 1 is 1 itself.**

**Step#5: Let’s start with a division table and write the number 1 in the quotient place as well as in the divisor place.**

**Step#6: Make the division considering,**

- Divisor 1
- Quotient 1
- Dividend 3

So, by the division method, 1 multiplied by 1 gives 1. Subtract 1 from 3, there will be remainder 2.

**Step#7: In the division table, 2 is the remainder. Here, we have to carry down 2 zeros after 2. So, it will become 200. The decimal point will come after 1 in the quotient.**

**Step#8: Now, we will add 1 in the divisor and make it 2.** **Step#9: The number next to 2 shall be selected in such a way that if we multiply the new combined number with the new number, then the value should be equal to 200 or less than that.** Now,

- If we take 6, the combination number will be 26. So, 26×6=156.
- If we take 7, the combination number will become 27. So, 27×7=189.
- If we take 8, the combination number will become 28. So, 28×8=224, which is more than 200.

Hence, the number 7 is acceptable and we get the value of 189. Now, we have to subtract 189 from 200, and we get the remainder of 11. So, we get 7 after the decimal.

**Step#10: As 11 is the remainder, we should carry down 2 zeros after 11. So, it will become 1100.** **Step#11: Now, we get 27, So, 27+7=34. We have to select another number next to 34, in such a way that if we multiply the new combined number with the new number, then the value should be equal to1100 or less than that.** Now,

- If we take 2, the combination number will be 342. So, 342×2=684.
- If we take 3, the combination number will become 343. So, 343×3=1029.
- If we take 4, the combination number will become 344. So, 344×4=1376, this is more than 1100.

Hence, if we take 3, the combination number will become 343. So, 343×3=1029, which is less than 1100. 3 will be added in the second decimal point. Hence, the number 3 is acceptable and we get 343. Now, we have to subtract 1029 from 1100, and we get the remainder 71.

**Step#12: As 71 is the remainder, we should carry down 2 zeros after 71. So, it will become 7100.** **Step#13: Now, we get 343, So, 343+3=346. We have to select another number next to 346, in such a way that if we multiply the new combined number with the new number, then the value should be equal to 7100 or less than that.** Now,

- If we take 2, the combination number will be 3462. So, 3462×2=6924.
- If we take 3, the combination number will become 3463. So, 3463×3=10389, this is more than 7100.

Hence, if we take 2, the combination number will become 3462. So, 3462×2=6924, which is less than 7100. 2 will be added in the third decimal point.

Hence, the number 2 is acceptable and we get 3462. Now, we have to subtract 6924 from 7100, and we get the remainder 76.

**Step#14: As 76 is the remainder, we should carry down 2 zeros after 76. So, it will become 7600.**

**Step#15: Now, we get 3462, So, 3462+2=3464. We have to select another number next to 3464, in such a way that if we multiply the new combined number with the new number, then the value should be equal to 7600 or less than that.** Now,

- If we take 0, the combination number will be 34640, which is more than 7600.

Hence, the number 0 shall be selected as 4 decimal point.

**Step#16: So, with the help of the above steps, we get the quotient as 1.7320, or 1.732 (we say)**

### Average Method for Square Root 3

In ancient times, the average method was used to find the square root of 3. This is very simple, and we get the value of the square root of 3 based on averages in a few steps. Let us see all steps,

**Step#1: We have to check the perfect squares just below 3 and above 3. In the case of number 3, 1 is the perfect square below 3, and 4 is the perfect square above 3.** 1 3 4

**Step#2: Find out the square root of the perfect square of both the numbers.** 1^{2} & 2^{2} So, square roots are 1 & 2.

**Step#3: Number 3 is between 1 & 4. Hence, naturally, the square root of 3 also will be between the square root of 1, i.e. 1, and the square root of 4, i.e. 2.** 1<3<4 1^{2}<3<2^{2} ^{or, }1<Square root of 3<4.

**Step#4: Divide the number 3 by any of one number, 1 or 2.** Let us divide 3 by 2, So, 3/2=1.5

**Step#5: Hence, we get a value of 1.5, and we will consider divisor 2 which was divided.**

**Now, we will calculate the average value of them,** Average value = (2+1.5)/2=3.5/2=1.75

**Step#6: In case, if you required further precise value, the process is to be continued. The above 1.75 value is almost closed, and we can select the same.**

### Equation Method for Square Root 3

In the method, you have to remember a simple equation. It is one of the simplest methods if you can remember the simple equation and within a few seconds only, you can calculate the square root of any digit. Equation is as below:

Check out our ANIMATED VIDEO on the Equation method!

Square root 3 equation method √(x±y) = √x ±y/(2√x) Perfect square to remember the square root

Here, x or y is the perfect square. It is better to remember all the basic perfect square table, Perfect square to remember

**Step#1: Split the digit into two numbers, in such a way that x can be written as a perfect square. ** We are going to calculate, the square root of 3.

Hence, 3 can be written as 4-1, since we know from the perfect square table, 4 is a perfect square, which means x=4 and y=1.

**Step#2: Input the value of x & y in the equation.**

√(x±y) = √x ±y/(2√x)

or, √(4-1) = √4 -1/(2√4)

or, √3 = 2 -1/(2×2)

or, √3 = 2 – 0.25 =1.75 (approx. value)

## Is Square 3 a Root Irrational or Rational?

There is no termination in the decimal part of 3 square roots. Irrational numbers have no decimal point at all. As we can see, the square root of 3 is never-ending in decimal form 1.7320508. This proves that the square root of 3 is not rational.

## How Do You Prove 3 is not a Perfect Square?

In order to calculate the square root of three, we should use the division method. The actual value of the square root of 3 is 1.732, so it can’t be factorized.

Adding up, square of square root 3 gives you the number 3. When multiplied by itself, it is a real number. In this instance, the square root of 3 is not a natural number, since it is a fraction and symbolized by √3. By multiplying the root by itself, you get the first number. As such, it is the foundation for the first or original number.

We’d like to determine first if the amount underneath the basis is a perfect square or not before calculating the root of any real. The prime factorization method can easily find the root of variety if it is a perfect square. As an example, the root of 4 is ± 2, since the square of 2 = 4. Here, however, 3 is not a square.

## Mathematic Application of Square Root 3

It can be approximated by taking the square root of 3, which is an irrational number

√3 ≈ 18817/10864 ≈ 1.7320508

Solution: The square root of 3 is considered an irrational number. It cannot be expressed as a decimal. The decimal expansion of p/q for integer’s p & q does not repeat nor terminate. You can express it by a continued fraction:

In some cases, the principal square root of 3 can also be thought of as this positive square root. By truncating the continued fraction early, we get rational approximations to V3. For Example:

## Uses of Square Root of 3

Square roots and radicals play a role in computation because they show up when we calculate areas, which is a very practical application. Let’s imagine renting a place one day. There are 400 square feet in this new apartment, which seems like a lot of space. It is especially annoying having been confined to a cupboard under the stairs for the past 11 years. This room must be 20 feet by 20 feet based on the square root.

## Square Root of 3 Solved Examples

### Over the course of two hours, James travels down the highway at an average speed of 100 √3 km/hr. How far does he travel each day?

It is necessary to use the formula Distance = Speed * Time to calculate distance

Speed = 100 √3 km/hr = 100 * 1.732 = 173.2

Time= 2 hour

Using the above formula, we get = 173.2 *2 = 346.4 kilometers.

This means that James covers an area of 346.4 kilometers.

### Rossi wanted to know whether -√3 is the same as √-3. How do you feel about this?

Real numbers cannot have negative square roots.

Numbers like – √3 are real numbers.

The number √ -3, however, is an imaginary one.

As a result, – √3 and √ -3 are not the same.

### Does the square root of 8 have an exponent form?

It is possible to convert all square roots using number bases with fractional exponents. In terms of 3’s square root, there is no exception. You can convert the square root of 3 to a base with an exponent using the following rule and answer.

√x = x ½

√3 = 3 ½

### If a circle has an area of 3π in2, then determine the circle’s radius?

We can define the circle’s radius as ‘r’.

A circle has an area of πr2 = 3π in2

r = ±√3 inches

A radius cannot be negative.

r = √3

There is 1.732 as the square root of 3.

r = 1.732 inches.

### What is the best way to calculate the square root of 8 on a computer?

When you have Excel or Numbers installed on your computer, you can get the square root of 3. To get the exact square root of 3, enter SQRT (3) into a cell. Following is the result obtained using 13 decimals. The square root of 3 is expressed in decimal form here.

SQRT (3) = 1.7320508089768

### What is the best way to calculate the square root of 8 using a calculator?

A calculator is one of the easiest and most boring methods for calculating the square root of 3. You can find the answer by typing the square root of 3 followed by root x. Your calculator generated the following result with nine decimal places.

√3= 1.732050808

### How to calculate (2√3 + 2 – √3) / (√3 -1)?

(2√3 + 2 – √3) / (√3 -1)

= (√3 +2) / (√3 -1)

= (√3 + 2) (√3 + 1) / (√3 -1) (√3 + 1)

= (√3 + 2) (√3 + 1) / 2

= 10.196/2

= 5.098

**Answer: 5.098**

### Can the square root of 3 be expressed as a fraction?

Since the square root of 3 is irrational, we can’t make an exact fraction. We can approximate the square root of 3 to a fraction by rounding it up to the nearest hundredth.

### How to calculate (15 √3 + 3√3) * (2√3 * 9√3)?

(15 √3 + 3√3) * (2√3 * 9√3)

= 18√3 * 54

=1683.55

**Answer:** 1683.55

### What are the merits of square root of 3?

The shape of a square root function resembles the steep increase at first, followed by saturation at the end. The square root transform inflates smaller numbers while stabilizing larger ones.

### How to calculate (37√3 -23√3)/7 + (55 √3 + 45 √3)/10?

(37√3 -23√3)/7 + (55 √3 + 45 √3)/10

=14√3 /7 + 100√3/10

= 2√3 + 10√3

= 12√3

= 12 * 1.732

= 20.784

**Answer:** 20.784

### How to calculate (√3 – 1)/ (√3 +1)?

(√3 – 1)/ (√3 +1)

= (√3- 1) (√3 – 1) / (√3 + 1) (√3 – 1)

= (√3 – 1) ² / (√3) ² – (1) ²

= 4 – 2 √3/ 3-1

= 2 (2 – √3) /2

= 2-√3

= 2- 1.732

= 0.268

**Answer:** 0.268

## Does the square root of 8 have the option of being rounded?

The square root of 3 should be rounded to the nearest tenth with one digit after the decimal point. If you round the square root of 3 to the nearest hundredth, you want two digits after the decimal point. Rounding the square root of 3 to the nearest thousandth requires three digits after the decimal point.

10^{th} Form = √3 = 1.7

100^{th} Form = √3 = 1.73

1000^{th} Form = √3 = 1.732

### How to calculate (√3/2 + 2√3)/5 + (7√3 – 4√3)/ √3?

(√3/2 + 2√3)/5 + (7√3 – 4√3)/ √3

= (√3 + 4√3)/10 + 3√3/√3

=5√3/10 + 3

=√3/2+ 3

= 0.866 + 3

=3.866

**Answer:** 3.866

### How to calculate (2√5 * 12√3 + 3√3 * 16) / (7 √3 * 9 – 3 √75)?

(2√5 * 12√3 + 3√3 * 16) / (7 √3 * 9 – 3 √75)

= 3√3 (2√5 * 4 + 16)/ 3√3 (7 * 3 – √25)

= (2√5 * 4 + 16) / (21 – 5)

= (8√5 + 16) / 16

= 8 (√5 + 2) / 16

= (√5 + 2) / 2

= 4.236/2

= 2.118

**Answer: 2.118**

## FAQs on Square Root of 3

**How do you find the square root of 3?**

When multiplied by itself, the square root of a number produces the original number. It is the square root of 25 that results from multiplying five times five.

Consequently, we have square roots of some numbers that do not affect whole numbers. Let’s take 3, for instance. Taking the square root of 3 in any form is easy. In the decimal form, it is denoted as 1.732. In the radical form, it is denoted as √3 in the exponent form, it is denoted as (3) ½.

**What is the value of √3?**

Approximately 1.732 is the value of square root of 3. Mathematicians often use this value. Because root 3 is an irrational number, it cannot be expressed using a fraction. Consequently, its decimals are infinite.

**How do you simply the square root of 3?**

Prime numbers include the number 3. Thus, the number 3 cannot be a power of 2, and is pair less. As a result, it is impossible to simplify the radical form of square root 3.

**What is the square of 3?**

The square of a number is calculated by multiplying it by itself. The square of three equals (3) ², the square of three equals nine.

**Is the square root of 3 irrational?**

It is an irrational number to square root 3 due to its irrationality. Theodorus of Cyrene, who proved its irrationality, also has the name Theodorus’ constant.

The number of non-repeating digits in irrational numbers is infinite after the decimal point. As symbols cannot be expressed in p*q, irrational numbers include square root of 2, 3, 5, and so on.

**Is root 3 a real number?**

It is true that √3 is a real number. A real number is composed of both rational and irrational elements. Due to its irrationality, we can also state that square root of 3 is a real number since it is irrational.

**Is square root of 3 an integer?**

It is important to note that the square roots of perfect squares (e.g., 0, 1, 4, 9, 16) are integers. In all other cases, positive integer square roots have non-repeating decimals in their decimal representations because they are irrational numbers. Therefore, the square root of 3 cannot be rounded up to a whole number. So square root of 3 is not an integer.

**What is √3 as a fraction?**

Using the radical symbol √, the square root of 3 can be written as √3. 1.732 is approximately the value of √3. Mathematicians use this value extensively. A fraction cannot represent root 3 since it is an irrational number.

**Does the square root of 3 have an infinite value?**

In the case of √3, it is an irrational number. A decimal number with an infinite number of occurrences is the answer.

**An expression with a square root of 3 means what?**

A cube root symbol is the radical symbol used for square roots with a little three to designate a cube root. The cube root of 27 equals three like we say that the cube root of 27 equals three.

**How long does it take for the square root of 3 to end?**

It is an irrational number to take the square root of 3. It cannot be expressed in the form pq for integers p,q, and its decimal expansion is neither repeated nor terminated.

**Why is the square root of 3 so important?**

If multiplied by itself by square root of 3, 3 is the answer which is a positive real number.

**Why do you divide voltage by square root 3?**

If we have a three-phase system, we multiply the phase voltage by the square root of 3. An equilateral triangle represents the three-phase system. The phase angle between each leg is 60 degrees.

**What is main purpose of square root?**

As a superset of rational numbers, algebraic numbers are mapped into square roots by the square root function. Euclidean norms and distance are defined by the square root of a nonnegative number, as are Hilbert spaces as generalizations.

## Conclusion

Hence, we have learned the square root of 3 along with different methods. Any doubt or if you have any suggestions, please let us know. You can refer to our most interesting articles

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