Square Root of 674

The square root is the inverse operation of squaring a number. 674 is not a perfect square. The square root of 674 is expressed in both radical and exponential forms. In the radical form, it is expressed as √674, whereas in the exponential form, it is expressed as (674)^(1/2). √674 ≈ 25.929, 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 typically used for perfect square numbers. However, for non-perfect square numbers like 674, methods such as the long-division method and approximation method are used. Let us explore these methods:

• Prime factorization method
• Long division method
• Approximation method

Prime factorization involves expressing a number as the product of its prime factors. However, 674 is not a perfect square, and its prime factors cannot be paired to simplify its square root. Thus, calculating √674 using prime factorization directly is not feasible.

The long division method is often used for calculating the square root of non-perfect square numbers. Below are the steps for using this method:

Step 1: Group the digits of 674 from right to left. For 674, group it as 74 and 6.

Step 2: Find the number whose square is less than or equal to 6. This number is 2, since 2 × 2 = 4. Subtract 4 from 6 to get a remainder of 2.

Step 3: Bring down 74 to make it 274 as the new dividend. Double the previous divisor (2) to get 4.

Step 4: Find a digit n such that 4n × n is less than or equal to 274. The appropriate n is 6, since 46 × 6 = 276, which is too large, so we use n = 5, giving 45 × 5 = 225.

Step 5: Subtract 225 from 274 to get 49.

Step 6: Since the remainder is less than the divisor, add a decimal point and bring down double zeros, making it 4900.

Step 7: Find the new divisor and continue the process to reach a further decimal precision.

The square root of 674 using the long division method is approximately 25.929.

The approximation method is another approach to finding the square root of a number. Here’s how to approximate the square root of 674:

Step 1: Identify the closest perfect squares to 674. The closest perfect squares are 625 (25²) and 676 (26²). √674 falls between 25 and 26.

Step 2: Use the formula: (Given number - smaller perfect square) / (Larger perfect square - smaller perfect square). (674 - 625) ÷ (676 - 625) = 0.96 Adding this to 25, we get 25 + 0.96 = 25.96.

Therefore, the approximate square root of 674 is 25.96.

• Square root: The square root is the inverse operation of squaring a number. For example, if 5² = 25, then √25 = 5.
   
• Irrational number: An irrational number cannot be expressed as a simple fraction. The decimal form of an irrational number is non-repeating and non-terminating.
   
• Principal square root: A number has both positive and negative square roots, but typically the positive square root is used, known as the principal square root.
   
• Approximation: This is a method used to find an estimated value that is near the actual value. For square roots, approximation methods help find values for non-perfect squares.
   
• Long division method: A systematic method of finding the square root of a number by dividing, multiplying, and subtracting in a structured manner, step by step.