Square Root of 813

The square root is the inverse of the square of the number. 813 is not a perfect square. The square root of 813 is expressed in both radical and exponential form. In the radical form, it is expressed as √813, whereas (813)^(1/2) in the exponential form. √813 ≈ 28.5147, 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, the prime factorization method is not used for non-perfect square numbers where 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 813 is broken down into its prime factors.

Step 1: Finding the prime factors of 813 Breaking it down, we get 3 × 271: 3^1 × 271^1

Step 2: Now we found out the prime factors of 813. Since 813 is not a perfect square, the digits of the number can’t be grouped in pairs.

Therefore, calculating 813 using prime factorization is not feasible for finding its square root.

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 813, we need to group it as 13 and 8.

Step 2: Now we need to find n whose square is ≤ 8. We can say n is '2' because 2 × 2 = 4, which is less than or equal to 8. Now the quotient is 2, and after subtracting 4 from 8, the remainder is 4.

Step 3: Bring down 13 to make the new dividend 413.

Step 4: Double the current quotient to get our new divisor which becomes 4. Now we have to find a digit x such that 4x × x is less than or equal to 413.

Step 5: Trying x = 9, we find 49 × 9 = 441, which is more than 413. Trying x = 8, 48 × 8 = 384, which works. The new remainder is 413 - 384 = 29.

Step 6: Since the dividend is less than the divisor, add a decimal point which allows us to add two zeroes to the dividend. The new dividend is 2900.

Step 7: Now find the new divisor, which is 576, because 2856 × 6 = 2856, which is close to 2900.

Step 8: Subtract 2856 from 2900, resulting in 44.

Step 9: Continue doing these steps until the desired precision is achieved. The quotient becomes approximately 28.514.

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 813 using the approximation method.

Step 1: Find the closest perfect squares around 813.

The smallest perfect square is 784 (28^2) and the largest perfect square is 841 (29^2). √813 falls between 28 and 29.

Step 2: Apply the formula: (Given number - smaller perfect square) / (Larger perfect square - smaller perfect square) Using the formula: (813 - 784) / (841 - 784) = 29/57 ≈ 0.5088 Adding this value to the smaller perfect square root gives us 28 + 0.5088 = 28.5088, so the square root of 813 is approximately 28.514.

• Square root: A square root is the inverse of a square. Example: 4^2 = 16, and the inverse of the square is the square root, which is √16 = 4.

• Irrational number: An irrational number cannot be written in the form of p/q, where q is not equal to zero and p and q are integers.

• Prime factorization: A way to express a number as the product of its prime factors. For example, 813 = 3 × 271.

• Long division method: A step-by-step procedure used to divide numbers and find square roots of non-perfect squares.

• Approximation method: A method used to find the approximate value of a square root by identifying the closest perfect squares around the given number.