Square Root of 598

The square root is the inverse of the square of a number. 598 is not a perfect square. The square root of 598 is expressed in both radical and exponential forms. In radical form, it is expressed as √598, whereas (598)^(1/2) in the exponential form. √598 ≈ 24.454, 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 598 is broken down into its prime factors: Step 1: Finding the prime factors of 598 Breaking it down, we get 2 x 13 x 23: 2^1 x 13^1 x 23^1 Step 2: Now we found out the prime factors of 598. The second step is to make pairs of those prime factors. Since 598 is not a perfect square, the digits of the number can’t be grouped in pairs. Therefore, calculating 598 using prime factorization is not feasible.

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 598, we need to group it as 98 and 5. Step 2: Now we need to find n whose square is closest to 5. We can say n as ‘2’ because 2 x 2 = 4 is lesser than or equal to 5. Now the quotient is 2, after subtracting 5 - 4 the remainder is 1. Step 3: Now let us bring down 98 which is the new dividend. Add the old divisor with the same number 2 + 2 = 4 which will be our new divisor. Step 4: The new divisor will be the sum of the dividend and quotient. Now we get 4n as the new divisor, we need to find the value of n. Step 5: The next step is finding 4n x n ≤ 198. Let us consider n as 4, now 4 x 4 x 4 = 176. Step 6: Subtract 198 from 176, the difference is 22, and the quotient is 24. 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 2200. Step 8: Now we need to find the new divisor that is 49 because 494 x 4 = 1976. Step 9: Subtracting 1976 from 2200 we get the result 224. Step 10: Now the quotient is 24.4. Step 11: Continue doing these steps until we get two numbers after the decimal point. Suppose if there is no decimal values continue till the remainder is zero. So the square root of √598 is approximately 24.45.

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 598 using the approximation method. Step 1: Now we have to find the closest perfect square of √598. The smallest perfect square less than 598 is 576 and the largest perfect square greater than 598 is 625. √598 falls somewhere between 24 and 25. 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 (598 - 576) ÷ (625-576) = 22/49 ≈ 0.449 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 24 + 0.449 = 24.449, so the square root of 598 is approximately 24.45.

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, 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. Prime factorization: The process of expressing a number as the product of prime numbers. Long division method: A method used for dividing larger numbers, useful for finding square roots of non-perfect squares. Approximation method: A technique for estimating the square root of a non-perfect square by comparing it to the nearest perfect squares.