Last updated on May 26th, 2025
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.
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.
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.
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.
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.
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.
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.
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.
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.
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
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.
Calculate √849 x 5.
145.688
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.
What will be the square root of (841 + 8)?
The square root is 29.14
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.
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.
Perimeter of the rectangle = 2 × (length + width).
Perimeter = 2 × (√849 + 38) = 2 × (29.1376 + 38) = 2 × 67.1376 = 134.28 units.
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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