Square Root of 374

The square root is the inverse of squaring a number. 374 is not a perfect square. The square root of 374 is expressed in both radical and exponential forms. In radical form, it is expressed as √374, whereas in exponential form it is expressed as (374)^(1/2). √374 ≈ 19.3391, which is an irrational number because it cannot be expressed as a fraction p/q, where p and q are integers and q ≠ 0.

The prime factorization method is used for perfect squares. However, for non-perfect squares, 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. Let us look at how 374 is broken down into its prime factors:

Step 1: Finding the prime factors of 374 Breaking it down, we get 2 x 11 x 17: 2^1 x 11^1 x 17^1

Step 2: Since 374 is not a perfect square, the digits of the number can’t be grouped into pairs.

Therefore, calculating √374 using prime factorization is impractical for exact values.

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, group the numbers from right to left. In the case of 374, we group it as 74 and 3.

Step 2: Find n whose square is less than or equal to 3. n is 1 because 1 x 1 = 1 which is less than 3. The quotient is 1, and the remainder is 2 after subtracting 1 from 3.

Step 3: Bring down 74 to make it 274, which is the new dividend. Add the old divisor with itself: 1 + 1 = 2, which will be our new divisor.

Step 4: Now we use 2n as the new divisor, and find n such that 2n x n ≤ 274. Consider n as 9, then 29 x 9 = 261.

Step 5: Subtract 261 from 274; the difference is 13. The quotient is 19.

Step 6: Since the dividend is less than the divisor, add a decimal point. Adding the decimal point allows us to add two zeroes to the dividend. The new dividend is 1300.

Step 7: Now find the new divisor: 193 because 193 x 6 = 1158.

Step 8: Subtract 1158 from 1300; the result is 142.

Step 9: Continue until we get two decimal places.

The square root of √374 is approximately 19.34.

The approximation method is another way to find square roots, and it is an easy method to find the square root of a given number. Here’s how to find the square root of 374 using the approximation method:

Step 1: Find the closest perfect squares around 374.

The smallest perfect square is 361 (19^2) and the largest is 400 (20^2).

So √374 falls between 19 and 20.

Step 2: Apply the formula:

(Given number - smallest perfect square) ÷ (Largest perfect square - smallest perfect square)

Using the formula:

(374 - 361) ÷ (400 - 361) = 13 ÷ 39 ≈ 0.333

Add the integer part to the decimal: 19 + 0.333 = 19.333.

So the square root of 374 is approximately 19.333.

• Square root: A square root is the inverse of squaring a number. Example: 4^2 = 16, and the inverse of the square is the square root: √16 = 4.
   
• Irrational number: An irrational number cannot be written in the form of p/q, where q is not zero, and p and q are integers.
   
• Principal square root: A number has both positive and negative square roots, but the positive square root is more prominent due to its uses in the real world. It is also known as the principal square root.
   
• Approximation: The process of finding a value close to the actual but not exact value, often used in calculating square roots of non-perfect squares.
   
• Long Division Method: A step-by-step method used for finding square roots of non-perfect squares, involving division and subtraction in sequential steps.