Square Root of 3 – Step by Step

Let’s learn square root of 3!

Square Root of 3 – Step by Step

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 the square root of 3. Is it possible at all?

Yes, it is possible!

Let’s discuss, square root of 3.

Square Root of 3

Let us try to learn, 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 we do normally in our calculations? Simply write √3.

square root 3
Fig.1 Square root 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.

What is the square root of 3?

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

Example of Square Root of 3

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,

  • 7x2 =3x2+12
  • Or, 4x2 =12
  • Or, x=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 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.

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

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 step 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. 

write 3 as 3.00000000
Write 3 as 3.00000000

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.

write 3.00000000 into 3 00 00 00 00
Write 3.00000000 into 3 00 00 00 00

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 = 12

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
square root 3 long division
Square root 3 long division

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.

square root 3 step 7
Square root 3 step 7

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.

square root 3 step 8
Square root 3 step 8

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.

square root 3 step 11
Square root 3 step 11

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.

square root 3 step 13 step
Square root 3 step 13 step

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

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.

12 & 22

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

12<3<22

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 to be continued. The above 1.75 value is almost closed, and we can select the same.

Equation Method

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

square root 3 equation method
Square root 3 equation method

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, that 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)

Conclusion

Hence, we have learned the square root of three along with different methods. Any doubt or if you any suggestions, please let us know.

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