Square Root of 651

The square root is the inverse of the square of a number. 651 is not a perfect square. The square root of 651 is expressed in both radical and exponential forms. In radical form, it is expressed as √651, whereas in exponential form it is (651)^(1/2). √651 ≈ 25.5147, 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 product of prime factors is the prime factorization of a number. Now let us look at how 651 is broken down into its prime factors:

Step 1: Finding the prime factors of 651 Breaking it down, we get 3 × 217: 3 × (7 × 31)

Step 2: Now we have found the prime factors of 651. Since 651 is not a perfect square, the digits of the number can’t be grouped in pairs.

Therefore, calculating √651 using prime factorization alone is not possible.

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

Step 1: To begin with, we group the numbers from right to left. For 651, group it as 51 and 6.

Step 2: Find n such that n² ≤ 6. We can say n = 2 because 2 × 2 = 4, which is less than 6. The quotient is 2 and the remainder is 2.

Step 3: Bring down 51, making the new dividend 251. Double the quotient (2), giving a new divisor of 4.

Step 4: Find n such that 4n × n ≤ 251. For n = 6, 46 × 6 = 276, which is more than 251. Trying n = 5, we have 45 × 5 = 225, which fits.

Step 5: Subtract 225 from 251, leaving a remainder of 26. The quotient is now 25.

Step 6: Since the remainder is less than the divisor, we add a decimal point and bring down zeros, making the new dividend 2600.

Step 7: Double the quotient (25), making the new divisor 50, and find n such that 50n × n ≤ 2600. Trying n = 5 gives 505 × 5 = 2525.

Step 8: Subtract 2525 from 2600, leaving the remainder 75. The quotient is now 25.5.

Step 9: Repeat the process until you achieve the desired precision.

The square root of 651 is approximately 25.5147.

The approximation method is another way to find square roots. It is an easy method to estimate the square root of a given number. Let us learn how to find the square root of 651 using the approximation method.

Step 1: Find the closest perfect squares to √651. The smallest perfect square less than 651 is 625 (√625 = 25), and the largest perfect square greater than 651 is 676 (√676 = 26). Thus, √651 falls between 25 and 26.

Step 2: Apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square) Using the formula: (651 - 625) / (676 - 625) = 26 / 51 ≈ 0.51 Add this result to the smaller perfect square root: 25 + 0.51 ≈ 25.51.

So the square root of 651 is approximately 25.51.

• Square root: A square root is the inverse of a square. Example: 4² = 16, and the inverse of the square is the square root, which is √16 = 4.
   
• Irrational number: An irrational number is a number that cannot be written in the form p/q, where q is not equal to zero and p and q are integers.
   
• Principal square root: A number has both positive and negative square roots; however, it is always the positive square root that has more prominence due to its uses in the real world. That is why it is also known as the principal square root.
   
• Prime factorization: Breaking down a number into its basic prime numbers that multiply together to equal the original number. Example: 651 = 3 × 7 × 31.
   
• Decimal: A number that includes a decimal point followed by digits representing a fraction, such as 25.5147.