Square Root of 798

The square root is the inverse of the square of the number. 798 is not a perfect square. The square root of 798 is expressed in both radical and exponential form. In the radical form, it is expressed as √798, whereas (798)^(1/2) in the exponential form. √798 ≈ 28.2512, 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 798 is broken down into its prime factors.

Step 1: Finding the prime factors of 798 Breaking it down, we get 2 x 3 x 7 x 19: 2¹ x 3¹ x 7¹ x 19¹

Step 2: Now that we found the prime factors of 798, the second step is to make pairs of those prime factors. Since 798 is not a perfect square, therefore the digits of the number can’t be grouped in pairs.

Therefore, calculating 798 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 798, we need to group it as 98 and 7.

Step 2: Now we need to find n whose square is less than or equal to 7. We can say n as ‘2’ because 2² = 4 is lesser than or equal to 7. Now the quotient is 2, and after subtracting 4 from 7, the remainder is 3.

Step 3: Now let us bring down 98, which is the new dividend. Add the old divisor with the same number 2 + 2 to get 4, which will be our new divisor.

Step 4: The new divisor will be 4n, and we need to find the value of n such that 4n × n ≤ 398.

Step 5: The next step is finding 4n × n ≤ 398. Let us consider n as 7, now 47 × 7 = 329.

Step 6: Subtract 329 from 398; the difference is 69, and 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 6900.

Step 8: Now we need to find the new divisor that is 545 because 545 × 5 = 2725.

Step 9: Subtracting 2725 from 6900, we get the result 4175.

Step 10: Now the quotient is 28.2

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

So the square root of √798 is approximately 28.25.

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 798 using the approximation method.

Step 1: Now we have to find the closest perfect squares to √798.

The smallest perfect square less than 798 is 784, and the largest perfect square greater than 798 is 841. √798 falls somewhere between 28 and 29.

Step 2: Now we need to apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square)

Using the formula (798 - 784) / (841 - 784) = 14 / 57 ≈ 0.246 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 + 0.246 = 28.246, so the square root of 798 is approximately 28.25.

• 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, 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 a positive square root that has more prominence due to its uses in the real world. That is the reason it is also known as a principal square root.

• Prime factorization: The expression of a number as the product of its prime factors is called prime factorization. Example: The prime factorization of 18 is 2 × 3 × 3.

• Decimal: If a number has a whole number and a fraction in a single number, then it is called a decimal. For example: 7.86, 8.65, and 9.42 are decimals.