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 307.
The square root is the inverse of the square of the number. 307 is not a perfect square. The square root of 307 is expressed in both radical and exponential form. In the radical form, it is expressed as √307, whereas (307)^(1/2) in exponential form. √307 ≈ 17.52142, 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 307 is broken down into its prime factors.
Step 1: Finding the prime factors of 307 Since 307 is a prime number, it cannot be broken down further into other prime factors.
Step 2: Now we found out the prime factors of 307. As 307 is not a perfect square, calculating 307 using prime factorization is not straightforward.
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 307, we need to group it as 07 and 3.
Step 2: Now we need to find n whose square is less than or equal to 3. We can say n as ‘1’ because 1 x 1 is less than or equal to 3. Now the quotient is 1, and after subtracting 1 from 3, the remainder is 2.
Step 3: Now let us bring down 07, making the new dividend 207. Add the old divisor with the same number 1 + 1; we get 2, which will be our new divisor.
Step 4: With the new divisor 2, find 2n such that 2n x n ≤ 207. Let us consider n as 7, then 27 x 7 = 189.
Step 5: Subtract 189 from 207; the difference is 18, and the quotient is 17.
Step 6: Since the dividend is less than the divisor, we need to add a decimal point. Adding the decimal point allows us to add two zeroes to the dividend. Now the new dividend is 1800.
Step 7: Now we need to find the new divisor by considering 345 because 345 x 5 = 1725.
Step 8: Subtracting 1725 from 1800, we get the result 75.
Step 9: Now the quotient is 17.5.
Step 10: Continue doing these steps until we get two numbers after the decimal point. Suppose if there is no decimal value, continue till the remainder is zero.
So the square root of √307 ≈ 17.52.
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 307 using the approximation method.
Step 1: Now we have to find the closest perfect square of √307. The smallest perfect square less than 307 is 289 (17²), and the largest perfect square greater than 307 is 324 (18²). √307 falls somewhere between 17 and 18.
Step 2: Now we need to apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Using the formula (307 - 289) / (324 - 289) = 18 / 35 ≈ 0.514. Using this formula, we identified the decimal point of our square root. The next step is adding the value we got initially to the decimal number, which is 17 + 0.514 ≈ 17.514, so the square root of 307 is approximately 17.514.
Students do make mistakes while finding the square root, like forgetting about the negative square root or skipping long division methods, etc. Now let us look at a few of those mistakes that students tend to make in detail.
Can you help Max find the area of a square box if its side length is given as √307?
The area of the square is approximately 307 square units.
The area of the square = side².
The side length is given as √307.
Area of the square = side² = √307 x √307 = 307.
Therefore, the area of the square box is approximately 307 square units.
A square-shaped garden measuring 307 square meters is built; if each of the sides is √307, what will be the square meters of half of the garden?
153.5 square meters
We can just divide the given area by 2 as the garden is square-shaped.
Dividing 307 by 2 gives us 153.5.
So half of the garden measures 153.5 square meters.
Calculate √307 x 5.
Approximately 87.6
The first step is to find the square root of 307, which is approximately 17.52.
The second step is to multiply 17.52 by 5.
So, 17.52 x 5 ≈ 87.6.
What will be the square root of (307 + 12)?
The square root is approximately 18.
To find the square root, we need to find the sum of (307 + 12). 307 + 12 = 319.
√319 ≈ 17.86, which rounds to approximately 18.
Therefore, the square root of (307 + 12) is approximately ±18.
Find the perimeter of the rectangle if its length ‘l’ is √307 units and the width ‘w’ is 20 units.
We find the perimeter of the rectangle as approximately 75.04 units.
Perimeter of the rectangle = 2 × (length + width).
Perimeter = 2 × (√307 + 20) ≈ 2 × (17.52 + 20) ≈ 2 × 37.52 ≈ 75.04 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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