Square Root of 376

The square root is the inverse of the square of a number. 376 is not a perfect square. The square root of 376 is expressed in both radical and exponential form. In the radical form, it is expressed as √376, whereas (376)^(1/2) in the exponential form. √376 ≈ 19.3907, 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 typically used for non-perfect square numbers, where the long-division method and approximation method are often employed. 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 376 is broken down into its prime factors:

Step 1: Finding the prime factors of 376 Breaking it down, we get 2 x 2 x 2 x 47: 2^3 x 47^1

Step 2: Now we found out the prime factors of 376. The second step is to make pairs of those prime factors. Since 376 is not a perfect square, the digits of the number can’t be grouped in pairs.

Therefore, calculating 376 using prime factorization does not yield an 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 376, we need to group it as 76 and 3.

Step 2: Now we need to find n whose square is equal to or less than 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 76, 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 will be 2n. Now we need to find the value of n such that 2n x n is less than or equal to 276. Let us consider n as 9, then 29 x 9 = 261.

Step 5: Subtract 261 from 276, the difference is 15, and the quotient is 19.

Step 6: Since the dividend is less than the divisor, we need to add a decimal point and bring down two zeros to the dividend. Now the new dividend is 1500.

Step 7: The new divisor will be 388. We need to find n such that 388n x n is less than or equal to 1500. Let us consider n as 3, then 388 x 3 = 1164.

Step 8: Subtract 1164 from 1500, we get 336. The quotient now is 19.3.

Step 9: Continue doing these steps until we get two numbers after the decimal point. Suppose if there is no decimal values, continue till the remainder is zero.

So the square root of √376 is approximately 19.39.

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 376 using the approximation method.

Step 1: Now we have to find the closest perfect square to √376.

The smallest perfect square less than 376 is 361 and the largest perfect square greater than 376 is 400.

√376 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).

Going by the formula (376 - 361) ÷ (400-361) = 15 ÷ 39 = 0.3846

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 + 0.3846 ≈ 19.3846, so the square root of 376 is approximately 19.39.

• Square root: A square root is the inverse of a square. Example: 4^2 = 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, 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, it is always the positive square root that has more prominence due to its uses in the real world. That is the reason it is also known as the principal square root.
   
• Prime factorization: The process of breaking down a number into its prime factors. For example, the prime factorization of 28 is 2 x 2 x 7.
   
• Approximation: The process of finding a value that is close enough to the right answer, usually with some thought or calculation involved.