Square Root of 683

The square root is the inverse of squaring a number. 683 is not a perfect square. The square root of 683 can be expressed in both radical and exponential forms. In radical form, it is expressed as √683, whereas in exponential form, it is (683)¹/². √683 ≈ 26.1108, which is an irrational number because it cannot be expressed as a ratio p/q, where p and q are integers and q ≠ 0.

The prime factorization method is suitable for perfect square numbers. However, for non-perfect squares like 683, methods like long division or approximation are used. Let's explore these methods: 

• Prime factorization method 
• Long division method 
• Approximation method

The prime factorization of a number is the product of its prime factors. Let's see how 683 is broken down into its prime factors.

Step 1: Finding the prime factors of 683 683 is a prime number itself, having no factors other than 1 and 683. Therefore, the prime factorization method is not applicable for finding its square root.

The long division method is particularly used for non-perfect square numbers. Let's learn how to find the square root using this method, step by step.

Step 1: Group the numbers from right to left. For 683, we consider 683 as a single group.

Step 2: Find the largest number whose square is less than or equal to 683. This number is 26, since 26² = 676, which is less than 683.

Step 3: Subtract 676 from 683, leaving a remainder of 7.

Step 4: Bring down pairs of zeros to the right of the remainder.

Step 5: Double the divisor (26) to get 52, and find a digit x such that 52x * x is less than or equal to 700. Here, x = 1 works, as 521 * 1 = 521.

Step 6: Subtract 521 from 700, leaving 179.

Step 7: Bring down another pair of zeros, making it 17900.

Step 8: Continue the division process to get the desired precision. Thus, √683 ≈ 26.11.

The approximation method is a simpler way to find square roots. Let's use it to find the square root of 683.

Step 1: Identify the perfect squares closest to 683. The perfect squares are 676 (26²) and 729 (27²). Thus, √683 is between 26 and 27.

Step 2: Use linear interpolation: (683 - 676) / (729 - 676) = 7 / 53 ≈ 0.132

So, √683 ≈ 26 + 0.132 = 26.132

• Square root: The square root is the inverse operation of squaring a number. For example, 4² = 16, and the square root of 16 is √16 = 4.

• Irrational number: An irrational number cannot be expressed as a simple fraction p/q, where p and q are integers, and q ≠ 0.

• Prime number: A prime number is a number greater than 1 that has no divisors other than 1 and itself.

• Linear interpolation: A method used to estimate values between two known values on a line or curve.

• Long division method: A step-by-step method used to divide numbers and find square roots, especially useful for non-perfect squares.