Square Root of 337

The square root is the inverse of the square of the number. 337 is not a perfect square. The square root of 337 is expressed in both radical and exponential form. In the radical form, it is expressed as √337, whereas (337)^(1/2) in the exponential form. √337 ≈ 18.35756, 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, for non-perfect square numbers like 337, the long-division method and approximation method are used. Let us now learn the following methods:

• Long division method
• Approximation 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 337, we need to group it as 37 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 × 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: Bring down 37, making it 237, which is the new dividend. Add the old divisor with the same number 1 + 1 to get 2, which will be our new divisor.

Step 4: The new divisor will be the sum of the dividend and quotient. Now we get 2n as the new divisor, we need to find the value of n.

Step 5: The next step is finding 2n × n ≤ 237. Let us consider n as 8, now 28 × 8 = 224.

Step 6: Subtract 224 from 237, the difference is 13, and the quotient is 18.

Step 7: 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 1300.

Step 8: Now we need to find the new divisor that is 183 because 183 × 7 = 1281.

Step 9: Subtracting 1281 from 1300, we get the result 19.

Step 10: Now the quotient is 18.3.

Step 11: Continue doing these steps until we get two numbers after the decimal point. If there are no decimal values, continue until the remainder is zero.

So the square root of √337 is approximately 18.36.

The approximation method is another method for finding the square roots, and 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 337 using the approximation method.

Step 1: Now we have to find the closest perfect squares of √337.

The smallest perfect square less than 337 is 324, and the largest perfect square greater than 337 is 361. √337 falls somewhere between 18 and 19.

Step 2: Now we need to apply the formula for approximation: (Given number - smaller perfect square) / (Larger perfect square - smaller perfect square).

Using the formula (337 - 324) / (361 - 324) = 13 / 37 ≈ 0.35135. Using the formula, we identified the decimal point of our square root. The next step is adding the value we got initially to the decimal number: 18 + 0.35135 ≈ 18.35, so the square root of 337 is approximately 18.35.

• Square Root: A square root of a number is a value that, when multiplied by itself, gives the number. Example: 4^2 = 16, and the inverse of the square is the square root, so √16 = 4.

• Irrational Number: An irrational number is a number that cannot be written in the form of p/q, where p and q are integers and q is not equal to zero.

• Principal Square Root: A number has both positive and negative square roots; however, it is always the positive square root that is referred to as the principal square root due to its use in real-world applications.

• Prime Number: A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself.

• Decimal: A decimal is a number that has a whole number and a fractional part separated by a decimal point, such as 7.86 or 18.36.