Square Root of 721

The square root is the inverse of squaring a number. 721 is not a perfect square. The square root of 721 is expressed in both radical and exponential form. In radical form, it is expressed as √721, whereas (721)^(1/2) in exponential form. √721 ≈ 26.85144, 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 typically used for perfect square numbers. However, for non-perfect square numbers, methods like the long division method and approximation method are used. Let us now learn the following methods: 

• Prime factorization method 
• Long division method 
• Approximation method

Prime factorization involves expressing a number as the product of its prime factors. However, since 721 is not a perfect square, finding its square root using prime factorization is not straightforward.

Step 1: Finding the prime factors of 721

Breaking it down, we get 7 × 103: 7^1 × 103^1

Step 2: Since 721 is not a perfect square, the digits of the number can’t be grouped in pairs that result in a perfect square. Therefore, calculating √721 using prime factorization is not possible.

The long division method is particularly used for non-perfect square numbers. This method involves estimating the square root of a number by dividing it into pairs of digits. Let us now learn how to find the square root using the long division method, step by step.

Step 1: Group the numbers from right to left. For 721, we have groups 72 and 1.

Step 2: Find a number whose square is less than or equal to 72. Let n be 8, since 8 × 8 = 64 ≤ 72. The quotient is 8, and the remainder is 72 - 64 = 8.

Step 3: Bring down 1, making the new dividend 81. Double the quotient, 8, to get 16, which is part of the new divisor.

Step 4: Find a number x such that (160 + x) × x ≤ 81. Let x be 0, since 160 × 0 = 0 ≤ 81.

Step 5: Since the remainder is still 81, add a decimal point and two zeroes, making the new dividend 8100.

Step 6: Using 1600 as the new divisor, find a digit y such that (1600 + y) × y ≤ 8100. Let y be 5, since 1605 × 5 = 8025 ≤ 8100.

Step 7: Subtract 8025 from 8100 to get a new remainder of 75. The quotient is now 26.85.

Step 8: Continue these steps until the desired precision is reached.

The approximation method is another way to find square roots. This method involves estimating the square root by identifying perfect squares closest to the number. Let us learn how to find the square root of 721 using the approximation method.

Step 1: Identify perfect squares around √721. The closest perfect square less than 721 is 676 (26^2), and the closest perfect square greater than 721 is 729 (27^2). √721 falls between 26 and 27.

Step 2: Use linear interpolation to approximate the square root. Using the formula: (Given number - smaller perfect square) / (larger perfect square - smaller perfect square) (721 - 676) / (729 - 676) = 45 / 53 ≈ 0.849 Adding this to the base of 26 gives 26 + 0.849 = 26.849, so the approximate square root of 721 is ≈ 26.85.

• Square root: A square root of a number is a value that, when multiplied by itself, gives the original number. Example: The square root of 25 is 5, as 5 × 5 = 25.

• Irrational number: An irrational number cannot be written as a simple fraction, meaning its decimal form is non-repeating and non-terminating. Example: √2 is irrational.

• Principal square root: The principal square root is the non-negative root of a number. For example, the principal square root of 9 is 3.

• Perfect square: A perfect square is a number that is the square of an integer. Example: 36 is a perfect square because it is 6^2.

• Long division method: A technique used to find the square root of a number by breaking it down into simpler steps, often used for non-perfect squares.