Square Root of 5 – Long Division, Average & Equation Method

Square root of 5! Is it possible to calculate? Many times, we surprised and shocked as well. We know, square root of 4, or 9 or 16 and we can do it instantly but the square root of 5! Is it difficult?

Let’s welcome to our interesting session, square root of 5.

Square Root of 5

Let us try to understand, square root of 5. If we calculate it in a calculator, we see a long list of irrational numbers and it’s not possible to remember or even write, as there is no ending of digits. So, what we do in mathematics? We simply write √5.

Many times, writing √5, will not be able to serve the purpose and its numeric valve to be considered. Here, we must calculate the numeric value of the square root of 5.

What exactly the square root of 5 is?

The square root of 5 is defined as a number when we multiply with the same number, gives the result of 5. It means,

√X x √X =X

It is written by √5.

So, as per definition, we can say,

√5x√5 =5

There is much research going on the numbers of digits after the decimal point in the square root of 5. Many types of research show even a billion digits in decimal places.

We have indicated, the value of the square root of 5 up to 20 decimal places,

√5 = 2.23606797749978969640

Example of Square Root of 5

There are so many examples, where we have to use, √5. Let us consider one simple equation to solve the value of x,

• 5x² =2x²+15
• Or, 3x² =15
• Or, x² =15/3=5
• Or, x = √5

We used to write,

• 1+√5
• 5+√5
• 2-√5

Or sometimes, we multiply,

• 5x√5=5√5
• √5x√5x√5=5√5
• 10x5x√5=50√5 and so on.

Hence, it is clear the whenever we are getting the square root of 5, we simply write √5. How, how do we specify the exact value of √5?

For example, if you need 10√5kg of sugar, how do you measure?

Yes! It is possible only when we can able to find out the numerical value of the square root of 5.

How to Find the Square Root of 5?

There is various method to find out the square root of 5.

• Long division method
• Average method
• Equation method

Long Division Method

Whenever we try to find out a square too of any number other than perfect square, we opt for this method. This method is a little long but easy. This method is also called a long division method as well. Let’s learn all the steps,

Step#1:Write the number 5 as 5.00000000, remember we are not doing any change of the value of 5, as both numerical values are the same.

Step#2: Order the zeros, like, 5.00 00 00 00 as both are the same and it will help in the division method.

Step#3: Now, try to find out the perfect square just below 5. If we analyze, 0,1,2,3 & 4 are numbers among below 5, and 4 is the perfect square.

As we know, 4 = 2×2 = 2²

Step#4: Find out the square root of the perfect square, below 5. Here, it is 4 and the square root of 4 is 2.

Step#5: Make a division table and write the number 2 in quotient place as well as in the divisor place.

Step#6: Make the division considering,

• Divisor 2
• Quotient 2
• Dividend 5

So, by the division method, 2 multiplied by 2 gives 4. Subtract 4 from 5, there will be remainder 1.

Step#7: As 1 is the remainder, we must carry down 2 zeros after 1. So, it will become 100. Next, carry down two zeros and write it down after 1. Simultaneously decimal point will come after 2 in the quotient.

Step#8: Now, we will add 2 in the divisor and make it 4.

Step#9: We have to select a number next to 4, in such a way that if we multiply the new combined number with the new number, then the value should be equal to100 or less than that.

Now,

• If we take 1, the combination number will be 41. So, 41×1=41.
• If we take 2, the combination number will become 42. So, 42×2=84.
• If we take 3, the combination number will become 43. So, 42×3=126, which is more than 100.

Hence, the number 2 is acceptable and we get 84. Now, we have to subtract 84 from 100, and we get the remainder 16. So, we get 2 after the decimal.

Step#10: As 16 is the remainder, we should carry down 2 zeros after 16. So, it will become 1600.

Step#11: Now, we get 42, So, 42+2=44. We have to select another number next to 44, in such a way that if we multiply the new combined number with the new number, then the value should be equal to1600 or less than that.

Now,

• If we take 1, the combination number will be 441. So, 441×1=441.
• If we take 2, the combination number will become 442. So, 442×2=884.
• If we take 3, the combination number will become 443. So, 443×3=1329.
• If we take 4, the combination number will become 444. So, 444×4=1776, this is more than 1600.

Hence, if we take 3, the combination number will become 443. So, 443×3=1329, which is less than 1600. 3 will be added in the second decimal point.

Hence, the number 3 is acceptable and we get 443. Now, we have to subtract 1329 from 1600, and we get the remainder 271.

Step#12: As 271 is the remainder, we should carry down 2 zeros after 271. So, it will become 27100.

Step#13: Now, we get 443, So, 443+3=446. We have to select another number next to 446, in such a way that if we multiply the new combined number with the new number, then the value should be equal to 27100 or less than that.

Now,

• If we take 4, the combination number will be 4464. So, 4464×4=17856.
• If we take 5, the combination number will become 4465. So, 4465×5=22325.
• If we take 6, the combination number will become 4466. So, 4466×3=26796.
• If we take 4, the combination number will become 4467. So, 4467×7=31269, this is more than 27100.

Hence, if we take 6, the combination number will become 4466. So, 4466×6=26796, which is less than 27100. 6 will be added in the third decimal point.

Hence, the number 6 is acceptable and we get 4466. Now, we have to subtract 26796 from 27100, and we get the remainder 304.

Step#14: As 304 is the remainder, we should carry down 2 zeros after 304. So, it will become 30400.

Step#15: Now, we get 4466, So, 4466+6=4472. We have to select another number next to 4472, in such a way that if we multiply the new combined number with the new number, then the value should be equal to 30400 or less than that.

Now,

• If we take 0, the combination number will be 44720, which is more than 30400.

Hence, the number 0 can be taken as 4 decimal points.

Step#16: So, with the help of the above steps, we get the quotient as 2.2360, or 2.236 (we say)

You can watch our VIDEO on it,

Average Method

In the ancient time itself, the average method was used to find the square root of 5. This is very simple, and we get the value of the square root of 5 based on averages in a few steps. Let us see all steps,

Step#1: We have to check the perfect squares just below 5 and above 5. In the case of number 5, 4 is the perfect square below 5, and 9 is the perfect square above 5.

4 5 9

Step#2: Find out the square root of the perfect square of both the numbers.

2² & 3²

So, square roots are 2 & 3.

Step#3: Number 5 is between 4 & 9. Hence, naturally, the square root of 5 also will be between the square root of 4, i.e. 2 and the square root of 9, i.e. 3.

4<5<9

2²<5<3²

^or, 2<Square root of 5<3.

Step#4: Divide the number 5 by any of one number, 2 or 3.

Let us divide 5 by 2,

So, 5/2=2.5

Step#5: Hence, we get a value of 2.5, and we will consider divisor 2 which was divided. Now, we will calculate the average value of them,

Average value = (2+2.5)/2=4.5/2=2.25

Step#6: In case, if you required further precise value, the process to be continued. The above 2.25 vale is almost closer, and we can select the same.

Equation Method

In the method, you have to remember a simple equation. It’s not very complex, but you remember, you can calculate the square root of any number in a few seconds.

Equation

√(x±y) = √x ±y/(2√x)

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

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 5. Hence, 5 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, √5 = 2 +1/(2×2)

or, √5 = 2 +0.25 =2.25 (approx value)

Conclusion

Hence, we have learned the square root of 5 along with different methods. Any doubt or if you any suggestions, please let us know.