Square Root of 519

The square root is the inverse of the square of a number. 519 is not a perfect square. The square root of 519 is expressed in both radical and exponential forms. In the radical form, it is expressed as √519, whereas in the exponential form it is expressed as (519)^(1/2). √519 ≈ 22.768, 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, for non-perfect square numbers, methods such as the long division method and approximation method are used. Let us now learn the following methods: -

1. Prime factorization method 
2. Long division method 
3. Approximation method

The product of prime factors is the prime factorization of a number. Now let us look at how 519 is broken down into its prime factors.

Step 1: Finding the prime factors of 519 Breaking it down, we get 3 x 173: 3¹ x 173¹

Step 2: Now we found out the prime factors of 519. The second step is to make pairs of those prime factors. Since 519 is not a perfect square, the digits of the number can’t be grouped in pairs.

Therefore, calculating 519 using prime factorization does not yield a simple 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 519, we need to group it as 19 and 5.

Step 2: Now we need to find n whose square is ≤ 5. We can say n is ‘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 bring down 19, making the new dividend 119. Add the old divisor with the same number 2 + 2 to get 4, which will be our new divisor.

Step 4: We now have 4n as the new divisor; we need to find the value of n. Step 5: The next step is finding 4n x n ≤ 119. Let's consider n as 2, now 4 x 2 x 2 = 16.

Step 6: Subtract 119 from 116, the difference is 3, and the quotient is 22.

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 300.

Step 8: Now we need to find the new divisor. Calculate 440 + n, where n = 6, because 446 x 6 = 2676.

Step 9: Subtracting 2676 from 3000, we get the result 324.

Step 10: Now the quotient is 22.7.

Step 11: Continue doing these steps until we get two numbers after the decimal point or until the remainder is zero.

So, the square root of √519 is approximately 22.77.

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 519 using the approximation method.

Step 1: Now we have to find the closest perfect squares of √519. The smallest perfect square less than 519 is 484, and the largest perfect square greater than 519 is 529. √519 falls somewhere between 22 and 23.

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

Using the formula (519 - 484) / (529 - 484) = 35 / 45 = 0.777 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 22 + 0.777 = 22.777, so the square root of 519 is approximately 22.777.

• Square root: A square root is a value that, when multiplied by itself, gives the original number. For example, 4² = 16, and the square root of 16 is √16 = 4.

• Irrational number: An irrational number is a number that cannot be expressed as a simple fraction, meaning it cannot be written as p/q, where q is not equal to zero and p and q are integers.

• Principal square root: The principal square root is the non-negative root of a number.

• Perfect square: A perfect square is a number that is the square of an integer. For example, 4, 9, 16, and 25 are perfect squares.

• Decimal: A decimal is a number that has a whole number and a fractional part separated by a decimal point, such as 7.86, 8.65, and 9.42.