Let’s explore square root of 8

The square root of 8! Don’t be surprised! It is possible with a few easy methods. We have already calculated the square root of perfect squares, like 4, 9, 16, 25, 36, and so on. But what about 8?

Let’s welcome to a simple session, square root of 8.

Let us try to explore the square root of 8, how it is calculated. If you calculate its value in an excel sheet or in a calculator, you will get a big list of irrational numbers with a decimal point. Now, do you think it can be remembered or even it can be written during a calculation! It’s really hard to do this.

So, what we write or consider in our mathematical calculation? We simply write √8.

Most of the time, writing a square root of 8 that is √8, is not recommended as it doesn’t give numerical value.

Hence, we need to calculate the numeric value of the square root of 8.

Fig.1 Square Root of 5

The square root of 8 is defined as a number when we multiply with the same number, gives the result of 8. It means,

√**P** x √**P** =**P**

It is written by √8.

So, as per definition, we can say,

√8x√8 =8

There are several experiments on the digits after the decimal point in the square root of 8. For many types, this experiment gives even a million digits after the decimal.

We have written, the value of the square root of 8 up to 15 decimal places,

**√8 = 2.828427124746190**

There may be thousands of examples, where we use, √8. Let us take, a very simple equation to solve the value of x,

- 11x
^{2}=2x^{2}+72 - Or, 9x
^{2}=72 - Or, x
^{2 }=72/9=8 - Or, x = √8

We used to write,

- 1+√8
- 3+√8
- 5-√8

Or sometimes, we multiply,

- 3x√8=3√8
- √8x√8x√8=8√8
- 19x2x√8=38√8 and so on.

Hence, it is very much clear that whenever we are getting the square root of 8 in our calculation, we simply write √8. So, how do we specify the exact value of √8?

For example, if your teacher asks you to bring 7√8 kg of sugar, how do you bring it without knowing the exact numerical value of the square root of 8?

Now, if you know the numerical value of √8, then it is simple! So, let try to understand how to find the square root of 8?

There are various processes to find out the square root of 5.

- Long division process
- Average process
- Equation process

Long division process is widely used to find out the square root of any non-perfect squares, like square root of 2 or square root of 5 or square root of 8.

As the process is little lengthy division process, it is called long division process.

Let’s explore the process step by step,

**Step#1: Write the number 8 as 8.00000000, remember we are simply changing the format, there is no change of the value of 8. **

Write 8 to 8.00000000

**Step#2: In the second step, let’s order the number like, 8.00 00 00 00 as the value of both number is same and it will help to understand the long division process better. **

Write 8.00000000 to 8.00 00 00 00

**Step#3: Now, the third step is to find out the perfect square just below 8. If we try to find out we get the flowing numbers, 0,1,2,3,4,5,6,7 and among all the number 4 is the perfect square.**

As we all know that, 4 = 2×2 = 2^{2}

**Step#4: Next step is to calculate the square root of the perfect square below 8. Here, it is 4 and 2 is the square root of 4.**

**Step#5: Let’s make a division table where 2 is the quotient and 2 is also a divisor. **

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

- Divisor 2
- Quotient 2
- Dividend 8

So, by the division method, 2 multiplied by 2 gives 4. Subtract 4 from 8, there will be remainder 4.

**Step#7: As 4 is the remainder, we must carry down 2 zeros after 4. So, it will become 400. From this division, 2 will be in the quotient.**

**Step#8: As per division rules, add 2 in the divisor. So, it will become 4.**

**Step#9: We have to select a number just next to 4, so that if we multiply the new combined number with the new number, then the value should be equal to 400 or less than that.**

Now,

- If we take 7, the combination number will be 47. So, 47×7=329.
- If we take 8, the combination number will become 48. So, 48×8=384.
- If we take 9, the combination number will become 49. So, 49×9=441, which is more than 400.

Hence, the number 8 is acceptable and we get 384. Now, we have to subtract 384 from 400, and we get the remainder 16. So, we get 8 in the quotient after the decimal point.

**Step#10: In the above division, the remainder is 16. Now, carry down 2 zeros after 16. So, it will be 1600.**

**Step#11: Now, we get 48, So, 48+8=56. We have to select another number next to 56, 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 561. So, 561×1=561.
- If we take 2, the combination number will become 562. So, 562×2=1124.
- If we take 3, the combination number will become 563. So, 563×3=1689.

Hence, if we take, the combination number will become 562. So, 562×2=1124, which is less than 1600. 2 will be added in the second decimal point.

Hence, the number 2 is acceptable and we get 1124. Now, we have to subtract 1124 from 1600, and we get the remainder 476.

Fig. 6 Square root 5 reminder 271

**Step#12: As 476 is the remainder, we should carry down 2 zeros after 476. So, it will become 47600.**

**Step#13: Now, we get 562, So, 562+2=564. We have to select another number next to 564, in such a way that if we multiply the new combined number with the new number, then the value should be equal to 47600 or less than that.**

Now,

- If we take 7, the combination number will be 5647. So, 5647×7=39529.
- If we take 8, the combination number will become 5648. So, 5648×8=45184.
- If we take 9, the combination number will become 5649. So, 5649×9=50841.

Hence, if we take 8, the combination number will become 5648. So, 5648×8=45184, which is less than 47600. 8 will be added in the quotient.

Hence, the number 8 is acceptable and we get 5648. Now, we have to subtract 45158 from 47600, and we get the remainder 2442.

Step#14: As 2442 is the remainder, we should carry down 2 zeros after 2442. So, it will become 244200.

Step#15: Now, we get 5648, So, 5648+8=5656. We have to select another number next to 5656, in such a way that if we multiply the new combined number with the new number, then the value should be equal to 244200 or less than that.

Step#16: Now if we consider the number 4 to place the right side of 5654, then it will become 56564.

As per the division process, we will get the value of 4.

If we continue the process, it will become 2.82842

To remember, we will consider 2.828

You can watch our VIDEO on it,

In the earlier time, the average method apart from long division process was used to find the square root of 8. This is another a simple process, and we can calculate the value of the square root of 8 very easy in a few steps. Let us learn all steps,

**Step#1: In the very first time, find out the perfect squares just below 8 and above 8. In the case of number 8, 4 is the perfect square below 8, and 9 is the perfect square above 8.**

4 8 9

**Step#2: Then we have to calculate the square root of both the numbers. It is so easy as both the numbers are perfect squares.**

2^{2} & 3^{2}

So, the square root of 4 & 9 is 2 & 3 respectively.

**Step#3: Number 8 is between 4 & 9. Hence, naturally, the square root of 8 exists between the square root of 4, and the square root of 9. **

4<8<9

2^{2}<8<3^{2}

^{or, }2<Square root of 8<3.

**Step#4: Divide the number 8 by any of one number, 2 or 3.**

Let us divide 8 by 3,

So, 8/3=2.666

**Step#5: Hence, we get a value of 2.666, and we will consider divisor 3 which was divided. Now, next step is to calculate the average value of them,**

Average value = (2.666+3.0)/2=5.666/2=2.833

**Step#6: In case, if you need to get more accuracy, this average method to be continued. The above 2.833 vale is almost near to the actual value, hence, we are considering the same.**

We have already learned the equation method, when we have calculated the square root of 3, now the same principle we need to put to find the square roof of 8 also.

Just need to remember a very simple equation to find out.

Let’s learn the equation!

Equation

√(x±y) = √x ±y/(2√x)

Here, x or y will be the perfect square and since right-hand side square root of x is used, then x should be perfect square in the above equation. Hoe, you have already learned the square root of all perfect squares.

Perfect square to remember

**Step#1: Any number can be written as the summation of subtraction of two numbers. Split the number into two numbers. Remember x should be written as a perfect square. **

We are going to calculate, the square root of 8. Hence, 8 can be written as 9 – 1, since we know from the perfect square table, 9 is a perfect square, which means x=9 and y=1.

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

√(x±y) = √x ±y/(2√x)

or, √(9 -1) = √9 -1/(2√9)

or, √8 = 3 -1/(2×3)

or, √8 = 3 +0.167 =2.833 (Approximate value)

Hence, we have learned the square root of 8 along with the main three methods. Any doubt or if you any suggestions, we are glad to get!.