Square Root of 639

The square root is the inverse operation of squaring a number. 639 is not a perfect square. The square root of 639 can be expressed in both radical and exponential forms. In radical form, it is expressed as √639, whereas in exponential form it is written as (639)^(1/2). The value of √639 is approximately 25.27, which is an irrational number because it cannot be expressed as a simple fraction p/q, where p and q are integers and q ≠ 0.

The prime factorization method is typically used for perfect squares. For non-perfect squares like 639, methods such as the long division method and approximation method are used. Let us now explore these methods:

• Prime factorization method
• Long division method
• Approximation method

Prime factorization involves expressing a number as a product of its prime factors. Let's break down 639 into its prime factors:

Step 1: Finding the prime factors of 639 Breaking it down, we get 3 x 3 x 71: 3^2 x 71

Step 2: We found the prime factors of 639. Since 639 is not a perfect square, the digits of the number cannot be grouped into pairs.

Therefore, calculating √639 using prime factorization does not yield an exact result.

The long division method is used for non-perfect squares. Here’s how to find the square root using this method, step by step:

Step 1: Group the digits of 639 from right to left. In this case, we have 39 and 6.

Step 2: Find n such that n^2 is less than or equal to 6. Here, n is 2 because 2^2 = 4, and 4 ≤ 6. The quotient is 2, and the remainder is 6 - 4 = 2.

Step 3: Bring down the next pair, 39, making the new dividend 239. Double the quotient (2), giving us 4, which will be part of our new divisor.

Step 4: Find a digit x such that 4x × x ≤ 239. Let x be 5, as 45 × 5 = 225, and 225 ≤ 239.

Step 5: Subtract 225 from 239 to get a remainder of 14. The quotient is now 25.

Step 6: Since the remainder is less than the new divisor, add a decimal point to the quotient. Bring down a pair of zeros to the dividend, making it 1400.

Step 7: Double the quotient (25) to get 50, then find x such that 50x × x ≤ 1400. Let x be 2, giving 502 × 2 = 1004.

Step 8: Subtract 1004 from 1400 to get 396. The quotient becomes 25.2.

Step 9: Continue this process until you reach the desired decimal precision.

So the square root of √639 is approximately 25.27.

The approximation method is a simpler approach to finding square roots. Let's find the square root of 639 using this method:

Step 1: Identify the nearest perfect squares around 639. Here, 625 (25^2) and 676 (26^2) are the closest.

Step 2: Apply the formula: \( \frac{639 - 625}{676 - 625} = \frac{14}{51} \approx 0.27 \) Using this formula, we find the decimal part of our square root. Add this to the integer part, giving 25 + 0.27 = 25.27.

Therefore, the square root of 639 is approximately 25.27.

• Square root: A square root is the inverse operation of squaring a number. For example, if 4^2 = 16, then √16 = 4.
   
• Irrational number: An irrational number cannot be expressed as a simple fraction of two integers. For example, √2 is irrational.
   
• Principal square root: The principal square root is the non-negative square root of a number. For instance, the principal square root of 16 is 4.
   
• Long division method: A technique used to find square roots of non-perfect squares by performing division operations iteratively.
   
• Approximation method: A method to estimate the square root of a number by comparing it to nearby perfect squares.