Square Root of 671

The square root is the inverse of the square of the number. 671 is not a perfect square. The square root of 671 is expressed in both radical and exponential form. In the radical form, it is expressed as √671, whereas (671)^(1/2) in the exponential form. √671 ≈ 25.896, 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, the prime factorization method is not typically used for non-perfect square numbers where the long-division method and approximation method are more applicable. Let us now learn the following methods:

• Prime factorization method
• Long division method
• Approximation method

The prime factorization of a number is the product of its prime factors. Now let us look at how 671 is broken down into its prime factors:

Step 1: Finding the prime factors of 671 671 is a prime number, so it cannot be broken down further into smaller prime factors.

Therefore, calculating 671 using the prime factorization method is not viable.

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: Begin by grouping the numbers from right to left. In the case of 671, we group it as 71 and 6.

Step 2: Now 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. The quotient is 2, with a remainder of 2.

Step 3: Bring down 71, making it the new dividend of 271. Add the old divisor with itself: 2 + 2 = 4, forming the new divisor.

Step 4: Find a digit n so that 4n × n ≤ 271. Choose n as 5: 45 × 5 = 225.

Step 5: Subtract 225 from 271, resulting in 46. The quotient becomes 25.

Step 6: Since the dividend is less than the divisor, add a decimal point and zeros to the dividend. The new dividend is 4600.

Step 7: Find a new digit n: 500n × n ≤ 4600. Choose n as 9: 509 × 9 = 4581.

Step 8: Subtract 4581 from 4600, giving 19. The quotient is 25.89.

Step 9: Continue these steps until you achieve the desired precision.

The square root of √671 is approximately 25.896.

The approximation method is another method for finding the 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 671 using the approximation method.

Step 1: Find the closest perfect squares to √671. The nearest perfect squares are 625 (25^2) and 676 (26^2). √671 falls between 25 and 26.

Step 2: Apply the formula: (Given number - smaller perfect square) / (Larger perfect square - smaller perfect square). (671 - 625) / (676 - 625) = 46 / 51 ≈ 0.902 Add this decimal to the smaller perfect square's root: 25 + 0.902 ≈ 25.902.

Thus, the square root of 671 is approximately 25.902.

• Square root: A square root is the inverse operation of squaring a number. For example, the square of 5 is 25, and the square root of 25 is 5.
   
• Irrational number: An irrational number cannot be expressed as a simple fraction. It has a non-repeating, non-terminating decimal expansion, such as √671.
   
• Prime number: A prime number is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers. For example, 671 is a prime number.
   
• Decimal: A decimal is a number that includes a decimal point, representing a fraction of a whole number. For example, 25.896 is a decimal.
   
• Long division method: A technique used to find the square root of non-perfect squares by dividing the number into pairs, estimating, and refining the quotient step by step.