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?
Students need to gain a basic understanding of square roots to tackle math and science problems. Working with square roots requires you to think about numbers in a slightly different way, because they ask “what number multiplied by itself gives the following result?”. You are well able to understand the laws of square roots and answer any question involving them, whether they involve direct calculation or just simplification. There are a few problems when you square root a number. Let’s welcome to a simple session, square root of 8.
What is Square Root of 8?
Square Root of 8 Basics
Before understanding the square root of 8, let’s try to understand the square root basics. A square root is an inverse of squaring a number. A square root is a direct opposite of multiplying by itself. In this case, the three squared is nine, so the square root of nine is three. The square root symbol means the square root of nine is three.
Almost all calculators have a “√” symbol that tells you how to take the square root of a number. The square root of every number is two. In addition to multiplying three by three, you can multiply three by three negative to get nine, and so on. A number’s negative square root can be ignored in many cases, but sometimes it is important to remember that every number has two roots.
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.
What exactly the square root of 8 is?
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
Example of Square Root of 8
There may be thousands of examples, where we use, √8. Let us take, a very simple equation to solve the value of x,
- 11x2 =2x2+72
- Or, 9x2 =72
- Or, x2 =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?
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
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 = 22
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.
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 as well.
Average Method
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.
22 & 32
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
22<8<32
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.
Equation Method
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!
√(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)
Can 8 be Square Rooted?
The square root of 8 is a whole number if it is not a perfect square in mathematics. Due to the fact that 8 is not a perfect square in this instance, as shown in the calculations below.
Is the Square Root of 8 Irrational or Rational?
Many people are curious about whether the square root of 8 is rational or irrational. A rational number can be written as a fraction, but an irrational number cannot. You can check whether 8 is a perfect square by looking at it. When it is, it is a rational number. A number that is not a perfect square is an irrational number. We already know that 8 is not a perfect square, so √8 is also an irrational number.
Can We write a Square Root of 8 in a Fraction?
We discussed that only rational numbers can be written as fractions, and that irrational numbers cannot. We cannot convert the square root of 8 into an exact fraction, since it is an irrational number. It can, however, be approximated by taking the square root of 8 and rounding it to the nearest hundredth.
Find a Square Root of 8 using Prime Factorization Method
The prime factors of a number can be used to estimate square roots, and the division method can be employed to estimate square roots. Whenever possible, we should use the prime factorization method in order to calculate the square root of a perfect square. In decimal, on the other hand, the division method must be used to calculate the square root of a number that is not a perfect square.
The Prime Factorization Method requires you to first express the number 8 as a product of its prime factors in order to get its value.
Now, 8 = 23, that is 2 x 2 x 2. Therefore,
√8 = √23 or √2 x √2 x √2
Or, √8 = 2 x √2
Or, √8 = 2√2
Now, we know that √2= 1.41421.
Therefore, √8 = 2 x 1.41421.
Or, √8 = 2.82842.
Estimation of Square Root of 8
Finding the range of a larger number allows you to more quickly estimate its square root. You may also have to determine the square root of x. As an example, let’s say n and m are in such a way that n2 < x <m2. It follows then that the square root of x is a number that lies between n and m. By doing this, it will be easier to calculate the square root of larger numbers.
Therefore, if we use the estimation method to find root 8, we will find that square of 2 = 4, and a square of 3 = 9. This means, therefore, that the root of 8 can be found between 2 and 3. Despite this, since the square of 3 equals 9 which is significantly bigger than 8, the root 8 value lies between 2.8 and 2.9. The exact square root of 8 is 2.82842712475. It is much closer to the estimate than our estimation.
Square Root of 8 Solved Examples
When a value is multiplied by itself, it becomes a square. As opposed to this, the square root of a number is a value which when multiplied by itself makes the original value. Hence, both methods work in the opposite direction. Let’s solve a few square root of 8 questions.
Q-1: John practices baseball in his backyard. He has an 8-square-foot area in his yard. Can you tell me the height of the gate?
Answer: A measure of the gate’s side length can be found by taking the square root of its area.
- Area = (length) ²
- (Area) ½ = Length
By using the rule of Square root we get, √8 = 2√2= 2*1.414= 2.828.
In this case, the gate has a side length of 2.828 feet.
Q-2: How you calculate the square root of 8 with the help of a computer?
Answer: You can get the square root of 8 if you have Excel or Numbers installed on your computer. After that, you can enter SQRT (8) in a cell to get the accurate result of the square root of 8. Using 13 decimals, we got the following result. This is the decimal form of the square root of 8.
Square Root of 8 = 2.8284271247463
Q-3: How you calculate square root of 8 with the help of a calculator?
Answer: You can calculate the square root of 8 with your calculator, which is the easiest and most boring method. The answer can be found by typing 8 followed by root x. The following result was obtained from on your calculator using 9 decimal places.
√8 = 2.828427127
Q-4: Is there any exponent form of square root of 8?
Answer: Number bases with fractional exponents can be used to convert all square roots. There is no exception when it comes to the square root of 8. The following rule and answer can be applied to converting the square root of 8 to a base with an exponent.
√a = a ½
√8 = 8 ½
Q-5: There is a square garden that Blake would like to fence. There are 8 square feet in the garden. What is the length of each side?
Answer: Area = (length) ²
(Area) ½ = Length
You will need to find the square root of 8 to find the side of a square garden. This is equal to √8 = 2√2. This means that the side length of the garden is 2√2.
Q-6: Is it possible to have fraction value of square root of 8?
Answer: We cannot make an exact fraction out of the square root of 8 due to its irrationality. By rounding up the square root of 8 to the nearest hundredth, we can approximate it to a fraction.
Q-7: How to calculate 27+ √72?
Answer: The square root of that 27 + √72 is
= 27 + √72
= 27 + √8 * √9
= 27 + √8 * 3
= 27 + 2√2 * 3
= 27 + 2.828 * 3
= 35.484
Q-8: In the following example, the area of a square is 8 in². Calculate the length of each side?
Answer: Square sides are numbered ‘a’, like side= 8
A square has an area of 8 square inches
If a = ±√8 then
A negative length cannot exist,
a = √8 = 2√2 = 2.828 inches.
Q-9: Is it possible to round of Square root of 8?
Answer: You want one digit after the decimal point if you round the square root of 8 to the nearest tenth. You want two digits after the decimal point if the square root of 8 is rounded to the nearest hundredth. You need three digits after the decimal point if you round the square root of 8 to the nearest thousandth.
- 10th Form = √8 = 2.8
- 100th Form = √8 = 2.83
- 1000th Form = √8 = 2.828
Q-10: How do you determine 17-3√8?
Answer: The square root of that 17- 3 √8 is
17- 3 √8
= 17 – 3 * 2√2
= 17 – 3* 2.828
= 17 – 8.484
= 8.516
Q-11: What is the result of square root of 8 multiply by 7?
Answer: We can write square root as √8 = 2√2= 2*1.414= 2.828.
So math solution would be,
√8 * 7
= 2√2 * 7
= 2.828 * 7
= 19.798
Q-12: Can you describe difference between square root & cube root of 8?
Answer: Cube root of 8 = 2 where we take one factor of 2 on the 8
Like this, 8 = 2* 2* 2
Square root of 8=2√2
The addition is inversed by subtraction, while multiplication is inversed by division. Squaring a number is inversely equivalent to find its square root. If you multiply the square root of 8 by itself, you get 8. The square root of 8 means we need to find a number whose square root is 8.
So the difference between cube root 2 & square root 2 will be,
= 2√2 – 2
= 2.828 – 2
= 0.828
FAQs for Square Root of 8
As a fraction, it is approximately 2.828, which is the square root of 8 in radical form, which is represented as √8.
In math, we do two operations: Take the square root of 8 using the radical sign √8, and square 8 using the radical sign ². The following two math rules yield the same result no matter how you do the math operations.
√x²=X
(√x) ² = X
In the light of these two rules, it can be concluded that the answer to this problem is 8. To prove that the answer is 8, we will show you different methods of calculating the square root of 8 squared.
To get our answer, we must first take the square root of 8 and then square the result.
√8 = 2.828
(2.828) ² = 8
Your next step is to do the same operation in reverse order, by taking square 8 first, then taking its square root.
8² = 64
√64= 8
A math problem such as the square root of 8 squared might also be interesting using this rule.
√x² = √x * √x
In addition, 8 squared are equal to 8 even when the squared root is squared.
√8² = √8 * √8
√8 * √8 = 2.828 * 2.828
2.828 * 2.828 = 8
You can write a root of 8 = 2√2= 2*1.414= 2.828.
You might also find yourself wondering whether a number like 8 is rational or irrational when you work with its roots. Numbers that are rational can be written as fractions and numbers that are irrational cannot.
Checking if a number is rational or irrational can be accomplished simply by finding out if it is a perfect square. If it is a perfect square, it is a rational number. If it is not a perfect square, it is an irrational number.
In radical form, the square root of 8 is represented as √8 which is also equal to two times two, which are approximately 2.828. On multiplying a number by itself, the square root gives the original number. Due to the non-perfect square nature of 8, the value is expressed in root form.
· Identify the GCF of the numerator and denominator.
· Calculate the GCF by dividing the numerator and denominator.
· The simplified fraction of the given fraction should be written.
The digits will be paired up by putting a bar above them, starting on the left. When multiplied by itself, find a number that produces a product less than or equal to 8. That means that the number is 2. If the divisor is 2, then we get quotient 2 and remainder 4.
Put a blank on the right of the divisor and double it. Identify a large digit to fill in the blank for the new quotient, such that when multiplied with the new quotient, the resultant product is less than or equal to the dividend. You can write the remainder while dividing the number. If you want more decimal places, repeat this process.
The radical 8 can be simplified by making it smaller inside the radical 8. This is what we call “simplifying a surd”. It is possible to simplify the square root of eight.
8 = 2√2
In order to provide you with an idea, the square of 2 is 4 and the square of 3 is 9. Between these two digits, 8’s square root lies. Due to the fact that 3 squared is 9, which is greater than 8, it is most likely to lie between 2.8 and 2.9. Therefore, 2.82842712475 is the square root of 8.
Conclusion
Hence, we have learned the square root of 8 along with the main three methods. You can see square root of 2, square root of 3, square root of 5 as well. Any doubt or if you any suggestions, we are glad to get!. You can refer to our most interesting articles
