Square Root of 826

The square root is the inverse of the square of a number. 826 is not a perfect square. The square root of 826 is expressed in both radical and exponential form. In the radical form, it is expressed as √826, whereas as 826^(1/2) in exponential form. √826 ≈ 28.744, 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 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 826 is broken down into its prime factors.

Step 1: Finding the prime factors of 826

Breaking it down, we get 2 x 413: 2^1 x 413^1

Step 2: Now we found out the prime factors of 826. The second step is to make pairs of those prime factors. Since 826 is not a perfect square, the digits of the number can’t be grouped in pairs. Therefore, calculating 826 using prime factorization is impossible.

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 826, we need to group it as 26 and 8.

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

Step 3: Now let us bring down 26, which makes the new dividend 426. Add the old divisor 2 with the same number 2 + 2 = 4, which will be our new divisor.

Step 4: The new divisor will be the sum of the quotient times 2. We need to find the value of n such that 4n × n ≤ 426.

Step 5: After checking, we find that 47 × 7 = 329.

Step 6: Subtract 329 from 426, and the difference is 97. The quotient is 27.

Step 7: Since the dividend is less than the divisor, we need to add a decimal point. Adding the decimal point allows us to add two zeroes to the dividend. Now the new dividend is 9700.

Step 8: Now we need to find the new divisor, which is 547, because 547 × 1 = 547.

Step 9: Subtracting 547 from 9700, we get the result 9153.

Step 10: Now the quotient is 28.7.

Step 11: Continue doing these steps until we get two numbers after the decimal point. Suppose there are no decimal values, continue till the remainder is zero.

So the square root of √826 is approximately 28.74.

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 826 using the approximation method.

Step 1: Now we have to find the closest perfect square of √826. The smallest perfect square less than 826 is 784, and the largest perfect square greater than 826 is 841. √826 falls somewhere between 28 and 29.

Step 2: Now we need to apply the formula that is (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Going by the formula (826 - 784) ÷ (841 - 784) = 42 ÷ 57 ≈ 0.74.

Using the formula, we identified the decimal point of our square root. The next step is adding the value we got initially to the decimal number, which is 28 + 0.74 = 28.74, so the square root of 826 is approximately 28.74.

• Square root: A square root is the inverse of a square. Example: 4² = 16, and the inverse of the square is the square root, that is √16 = 4.
   
• Irrational number: An irrational number is a number that cannot be written in the form of p/q, 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 the reason it is also known as the principal square root.
   
• Prime factorization: The expression of a number as the product of its prime factors is called prime factorization. Long
   
• Division Method: A systematic method used to find the square root of non-perfect squares by dividing the number into pairs from right to left and using trial and error to find the quotient.