Square Root of 377

The square root is the inverse of the square of a number. 377 is not a perfect square. The square root of 377 is expressed in both radical and exponential form. In the radical form, it is expressed as √377, whereas in exponential form it is expressed as 377^(1/2). √377 ≈ 19.4165, 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:

• 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 377 is broken down into its prime factors.

Step 1: Finding the prime factors of 377 Breaking it down, we get 13 x 29.

Step 2: Now we found out the prime factors of 377. Since 377 is not a perfect square, the digits of the number can’t be grouped in pairs.

Therefore, calculating 377 using prime factorization is impossible.

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 377, we need to group it as 77 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 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 77, which is the new dividend. Add the old divisor with the same number 1 + 1, we get 2, which will be our new divisor.

Step 4: The new divisor is 20 (2n), and we need to find the value of n such that 20n x n is less than or equal to 277. Let us consider n as 1, now 20 x 1 x 1 = 201.

Step 5: Subtract 201 from 277; the difference is 76, and 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 7600.

Step 7: Now we need to find the new divisor, which is 391, because 391 x 1 = 391.

Step 8: Subtracting 391 from 7600, we get the result 7209. Step 9: Continue doing these steps until we get two numbers after the decimal point. Suppose if there are no decimal values, continue till the remainder is zero.

So the square root of √377 is approximately 19.42.

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 377 using the approximation method.

Step 1: Now we have to find the closest perfect squares of √377.

The smallest perfect square less than 377 is 361, and the largest perfect square greater than 377 is 400.

√377 falls somewhere between 19 and 20.

Step 2: Now we need to apply the formula:

(Given number - smallest perfect square) ÷ (Greater perfect square - smallest perfect square).

Using the formula, (377 - 361) ÷ (400 - 361) = 16 ÷ 39 ≈ 0.41.

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, which is 19 + 0.41 = 19.41, so the square root of 377 is approximately 19.41.

• Square root: A square root is the inverse of a square. Example: 4² = 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 prominent due to its uses in the real world. That is the reason it is also known as the principal square root.
   
• Prime factorization: Expressing a number as the product of its prime factors. For example, the prime factorization of 377 is 13 x 29.
   
• Long division method: A method used to find more precise square roots of non-perfect squares by dividing the number into pairs of digits, starting from the decimal point.