Square Root of 653

The square root is the inverse of the square of the number. 653 is not a perfect square. The square root of 653 is expressed in both radical and exponential form. In the radical form, it is expressed as √653, whereas (653)^(1/2) in the exponential form. √653 ≈ 25.549, 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 used for perfect square numbers. However, for non-perfect square numbers, 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 prime factorization of a number involves expressing it as a product of prime factors. However, as 653 is a non-perfect square and also a prime number, it cannot be broken down into smaller prime factors. Therefore, calculating the square root of 653 using prime factorization is not feasible.

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, we need to group the numbers from right to left. In the case of 653, we need to group it as 53 and 6.

Step 2: Now we need to find n whose square is less than or equal to 6. We can say n is ‘2’ because 2 × 2 = 4, which is less than 6. Subtracting, we get a remainder of 2.

Step 3: Bring down 53 to make the new dividend 253. Add the old divisor with the same number 2 + 2 = 4, which will be our new divisor.

Step 4: The new divisor is 4n. We need to find n such that 4n × n ≤ 253. Let us consider n as 5, now 45 × 5 = 225.

Step 5: Subtract 225 from 253, the difference is 28, and the quotient is 25.

Step 6: Since the dividend is less than the divisor, we need to add a decimal point. Adding the decimal point allows us to add two zeros to the dividend. Now the new dividend is 2800.

Step 7: Now we need to find the new divisor which is 510 because 510 × 5 = 2550.

Step 8: Subtracting 2550 from 2800, we get the result 250.

Step 9: The quotient is 25.5. Step 10: Continue this method to find more decimal places of the square root.

So the square root of √653 is approximately 25.549.

The approximation method is another method for finding square roots. It is an easy method to find the square root of a given number. Now let us learn how to find the square root of 653 using the approximation method.

Step 1: Find the closest perfect square of √653. The smallest perfect square less than 653 is 625 and the largest perfect square more than 653 is 676. √653 falls somewhere between 25 and 26.

Step 2: Apply the formula (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Using the formula (653 - 625) / (676 - 625) = 28 / 51 ≈ 0.549. Adding this to 25 gives us the approximate square root: 25 + 0.549 = 25.549.

Therefore, the square root of 653 is approximately 25.549.

• Square root: A square root is the inverse of a square. For example, 4² = 16, and the inverse is the square root, √16 = 4.
   
• Irrational number: An irrational number cannot be expressed as a fraction where the numerator and the denominator are integers, and the denominator is not zero.
   
• Principal square root: The principal square root is the non-negative square root of a number. It is the positive value of the two possible square roots of a number.
   
• Prime number: A prime number is a number greater than 1 that has no positive divisors other than 1 and itself.
   
• Decimal: A decimal is a way of representing numbers that have a whole number part and a fractional part separated by a decimal point. Examples include 7.86, 8.65, and 9.42.