Square Root of 373

The square root is the inverse of the square of the number. 373 is not a perfect square. The square root of 373 is expressed in both radical and exponential form. In the radical form, it is expressed as √373, whereas in exponential form, it is expressed as (373)^(1/2). √373 ≈ 19.3132, 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, the prime factorization method is not applicable, and methods such as long division and approximation are used. Let us now learn the following methods:

• Prime factorization method
• Long division method
• Approximation method

The product of prime factors is the prime factorization of a number. Now let us look at how 373 is broken down into its prime factors.

Step 1: Finding the prime factors of 373 Breaking it down, we find that 373 is a prime number

Therefore, it cannot be expressed as a product of other prime numbers.

Hence, the prime factorization method is not applicable for calculating the square root of 373.

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 373, we need to group it as 73 and 3.

Step 2: Now we need to find n whose square is less than or equal to 3. We can say n is '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 73, making the new dividend 273. 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 2n. We need to find the value of n such that 2n × n ≤ 273. Let's try n as 9. Now, 29 × 9 = 261.

Step 5: Subtract 261 from 273, and the difference is 12. The quotient is 19.

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 1200.

Step 7: Find the next divisor which is 193, because 193 × 6 = 1158.

Step 8: Subtract 1158 from 1200 to get 42 as the remainder.

Step 9: The quotient is now 19.3. Continue this process until you achieve the desired level of precision.

So the square root of √373 ≈ 19.313.

The approximation method is another method for finding the 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 373 using the approximation method.

Step 1: Find the closest perfect squares around 373. The smallest perfect square less than 373 is 361, and the largest perfect square greater than 373 is 400. √373 falls somewhere between 19 and 20.

Step 2: Apply the formula:

(Given number - smallest perfect square) / (Greater perfect square - smallest perfect square)

(373 - 361) / (400 - 361) = 12/39 ≈ 0.308

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 19 + 0.308 ≈ 19.308.

So the square root of 373 is approximately 19.308.

• Square root: The square root of a number is a value that, when multiplied by itself, gives the original number. Example: The square root of 16 is 4, since 4 × 4 = 16.
   
• Prime number: A prime number is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers.
   
• Irrational number: An irrational number is a number that cannot be expressed as a fraction for any integers, having an infinite and non-repeating decimal expansion.
   
• Long division method: A technique used to find the square root of non-perfect squares by dividing the number into groups and determining the quotient step by step.
   
• Approximation method: A method used to estimate the square root of a number by determining its position between two closest perfect squares and interpolating the decimal part.