129 LearnersLast updated on May 26th, 2025
Square Root of 849
If a number is multiplied by the same number, the result is a square. The inverse of the square is a square root. The square root is used in the field of vehicle design, finance, etc. Here, we will discuss the square root of 849.
What is the Square Root of 849?
The square root is the inverse of the square of the number. 849 is not a perfect square. The square root of 849 is expressed in both radical and exponential form. In the radical form, it is expressed as √849, whereas (849)^(1/2) in the exponential form. √849 ≈ 29.1376, which is an irrational number because it cannot be expressed in the form of p/q, where p and q are integers and q ≠ 0.
Finding the Square Root of 849
The prime factorization method is used for perfect square numbers. However, the prime factorization method is not used for non-perfect square numbers where the long-division method and approximation method are used. Let us now learn the following methods.
- Prime factorization method
- Long division method
- Approximation method
Square Root of 849 by Prime Factorization Method
The product of prime factors is the prime factorization of a number. Now let us look at how 849 is broken down into its prime factors.
Step 1: Finding the prime factors of 849
Breaking it down, we get 3 x 283: 3^1 x 283^1
Step 2: Now we found out the prime factors of 849. The second step is to make pairs of those prime factors. Since 849 is not a perfect square, therefore the digits of the number can’t be grouped in pairs. Therefore, calculating 849 using prime factorization is not feasible for finding the exact square root.
Square Root of 849 by Long Division Method
The long division method is particularly used for non-perfect square numbers. In this method, we should check the closest perfect square number for the given number. Let us now learn how to find the square root using the long division method, step by step.
Step 1: To begin with, we need to group the numbers from right to left. In the case of 849, we need to group it as 49 and 8.
Step 2: Now we need to find n whose square is 8 or less. We can say n as '2' because 2 x 2 = 4, which is less than 8. Now the quotient is 2, and after subtracting 4 from 8, the remainder is 4.
Step 3: Now let us bring down 49, making the new dividend 449. Add the old divisor, 2, to itself, getting 4, which will be our new divisor.
Step 4: Find n such that 4n x n is less than or equal to 449. Let's consider n as 9, now 49 x 9 = 441.
Step 5: Subtract 441 from 449, getting a remainder of 8. The quotient is now 29.
Step 6: Since the dividend is less than the divisor, we need to add a decimal point. Adding the decimal point allows us to bring two zeroes down to the dividend. Now the new dividend is 800.
Step 7: Now we need to find the new divisor, which is 58, because 581 x 1 = 581.
Step 8: Subtracting 581 from 800 gives us 219.
Step 9: Continue this process until the desired precision is achieved.
So the square root of √849 is approximately 29.14.
Square Root of 849 by Approximation Method
The approximation method is another method for finding square roots; it is an easy method to find the square root of a given number. Now let us learn how to find the square root of 849 using the approximation method.
Step 1: Now we have to find the closest perfect square of √849.
The smallest perfect square of 849 is 841, and the largest perfect square is 900. √849 falls somewhere between 29 and 30.
Step 2: Now we need to apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square) Using the formula (849 - 841) / (900 - 841) = 8 / 59 ≈ 0.1356
Using this approximation, the square root of 849 can be estimated as 29 + 0.1356 = 29.1356, so the square root of 849 is approximately 29.14.
Common Mistakes and How to Avoid Them in the Square Root of 849
Students do make mistakes while finding the square root, such as forgetting about the negative square root or skipping steps in the long division method. Now let us look at a few of those mistakes students tend to make in detail.
Mistake 1
Forgetting about the negative square root
It is important to make students aware that a number does have both positive and negative square roots. However, we will be taking only the positive square root, as it is the required one.
For example, √50 = 7.07, but there is also -7.07, which should not be forgotten.
Mistake 2
Not adding the square root symbol in proper places
Not using the square root properly is another mistake that children make. To resolve this, we need to teach children that simplifying the numbers inside the square root is important before moving on to the second step.
For example, √(12+24) = √12 + √24 is the wrong method of using roots. The right way is √(12+24) = √36.
Mistake 3
Not finding the correct value of a non-perfect square
The first step that needs to be taught to children is to simplify the numbers inside the square root. This is important so that students do not make mistakes while identifying the approximate value.
For example, √50 = 7 is wrong; the correct answer is √50 ≈ 7.071067.
Mistake 4
Confusing the square root symbol with the cube root
Students need to be taught the difference between cube root and square root constantly, as this will help them avoid mistakes.
For example, √50 and ∛50 are different.
Mistake 5
Making mistakes in long division
Finding a square root using the long division method involves many steps. Due to this, many students might skip a step accidentally, resulting in a wrong answer at the end of the sum. This may happen during subtraction or elsewhere.
Square Root of 849 Examples
Problem 1
Can you help Max find the area of a square box if its side length is given as √849?
The area of the square is 849 square units.
Explanation
The area of the square = side^2.
The side length is given as √849.
Area of the square = side^2 = √849 x √849 = 849.
Therefore, the area of the square box is 849 square units.
Problem 2
A square-shaped building measuring 849 square feet is built; if each of the sides is √849, what will be the square feet of half of the building?
424.5 square feet
Explanation
We can just divide the given area by 2 as the building is square-shaped.
Dividing 849 by 2 = 424.5.
So half of the building measures 424.5 square feet.
Problem 3
Calculate √849 x 5.
145.688
Explanation
The first step is to find the square root of 849, which is approximately 29.1376.
The second step is to multiply 29.1376 by 5.
So 29.1376 x 5 = 145.688.
Problem 4
What will be the square root of (841 + 8)?
The square root is 29.14
Explanation
To find the square root, we need to find the sum of (841 + 8), 841 + 8 = 849, and then find √849 ≈ 29.14.
Therefore, the square root of (841 + 8) is approximately ±29.14.
Problem 5
Find the perimeter of a rectangle if its length ‘l’ is √849 units and the width ‘w’ is 38 units.
The perimeter of the rectangle is approximately 134.28 units.
Explanation
Perimeter of the rectangle = 2 × (length + width).
Perimeter = 2 × (√849 + 38) = 2 × (29.1376 + 38) = 2 × 67.1376 = 134.28 units.
FAQ on Square Root of 849
1.What is √849 in its simplest form?
2.Mention the factors of 849.
3.Calculate the square of 849.
4.Is 849 a prime number?
5.849 is divisible by?
6.How does learning Algebra help students in United States make better decisions in daily life?
7.How can cultural or local activities in United States support learning Algebra topics such as Square Root of 849?
8.How do technology and digital tools in United States support learning Algebra and Square Root of 849?
9.Does learning Algebra support future career opportunities for students in United States?
Important Glossaries for the Square Root of 849
- Square root: A square root is the inverse of a square. Example: 4^2 = 16, and the inverse of the square is the square root, that is √16 = 4.
- Irrational number: An irrational number is a number that cannot be written in the form of p/q, where q is not equal to zero and p and q are integers.
- Principal square root: A number has both positive and negative square roots; however, the positive square root is more commonly used due to its applications in the real world. This is why it is known as the principal square root.
- Prime factorization: The process of expressing a number as the product of its prime factors. Example: The prime factorization of 849 is 3 x 283.
- Approximation: A method of finding an estimate that is close to the actual value, often used when the exact value is difficult to calculate.
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Jaskaran Singh Saluja
About the Author
Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.
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