Square Root of 652

The square root is the inverse operation of squaring a number. 652 is not a perfect square. The square root of 652 can be expressed in both radical form and exponential form. In radical form, it is expressed as √652, and in exponential form as \(652^{1/2}\). The value of √652 is approximately 25.529, which is an irrational number as it cannot be expressed as a fraction of two integers.

The prime factorization method is typically used for perfect square numbers. However, for non-perfect squares, methods like long division and 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 break down 652 into its prime factors:

Step 1: Finding the prime factors of 652 Breaking it down, we get 2 x 2 x 163: (2^2 times 163^1)

Step 2: Since 652 is not a perfect square, we cannot group all the prime factors into pairs.

Therefore, calculating √652 using prime factorization alone is not feasible.

The long division method is useful for finding the square root of non-perfect squares. Here is how to use this method step-by-step:

Step 1: Group the digits of 652 from right to left as '52' and '6'.

Step 2: Find the largest number whose square is less than or equal to 6. This number is 2, since (2^2 = 4). Subtract 4 from 6, leaving a remainder of 2.

Step 3: Bring down the next pair, 52, to get the new dividend, 252. Double the divisor from step 2 (2), giving us 4, and append a digit to form a new divisor.

Step 4: Determine the largest digit, n, so that 4n x n ≤ 252. Here, n = 5, since 45 x 5 = 225.

Step 5: Subtract 225 from 252, resulting in a remainder of 27. Bring down two zeros to form the new dividend, 2700.

Step 6: Repeat the process until the desired precision is achieved. Our quotient so far is approximately 25.52.

The approximation method provides a quick way to estimate square roots. Here's how to find √652 using approximation:

Step 1: Identify perfect squares nearest to 652. The closest perfect squares are 625 (25²) and 676 (26²). Therefore, √652 is between 25 and 26.

Step 2: Use the formula: ((text{Given number} - text{Lower perfect square}) / (text{Higher perfect square} - text{Lower perfect square})). Calculating, we get: ((652 - 625) / (676 - 625) = 27 / 51 approx 0.529).

Step 3: Add this decimal to the lower square root: 25 + 0.529 = 25.529.

Hence, the approximate value of √652 is 25.529.

• Square root: A square root of a number is a value that, when multiplied by itself, gives the original number. Example: \(5^2 = 25\), and the square root of 25 is √25 = 5.
   
• Irrational number: An irrational number cannot be expressed as a simple fraction. Its decimal goes on forever without repeating.
   
• Perfect square: A perfect square is an integer that is the square of an integer. For example, 25 is a perfect square because it is (5^2).
   
• Long division method: A technique used to find the square root of numbers that are not perfect squares.
   
• Approximation: A method to estimate the value of a mathematical expression, often using nearby or simpler values.