Square Root of 518

The square root is the inverse of the square of the number. 518 is not a perfect square. The square root of 518 is expressed in both radical and exponential form. In the radical form, it is expressed as √518, whereas (518)^(1/2) in the exponential form. √518 ≈ 22.748, 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:

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 518 is broken down into its prime factors:

Step 1: Finding the prime factors of 518 Breaking it down, we get 2 x 259, and 259 is further factored into 7 x 37. So, the prime factorization of 518 is 2 x 7 x 37.

Step 2: Now we have the prime factors of 518. Since 518 is not a perfect square, the digits of the number can’t be grouped in pairs.

Therefore, calculating the square root of 518 using prime factorization is not feasible.

The long division method is particularly used for non-perfect square numbers. 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 518, we need to group it as 18 and 5.

Step 2: Now we need to find n whose square is less than or equal to 5. We can say n is ‘2’ because 2 x 2 = 4, which is less than 5. The quotient is 2, and after subtracting, the remainder is 1.

Step 3: Bring down 18, making the new dividend 118. Add the old divisor with the same number, 2 + 2, which gives us 4 as the new divisor.

Step 4: The new divisor will be 4n. We need to find the value of n such that 4n x n ≤ 118. Considering n as 2, we have 42 x 2 = 84, which is less than 118.

Step 5: Subtract 84 from 118; the difference is 34, and the quotient is 22.

Step 6: 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 3400.

Step 7: Find the new divisor. It is 449 because 449 x 7 = 3143.

Step 8: Subtracting 3143 from 3400, we get 257.

Step 9: Now the quotient is 22.7.

Step 10: Continue these steps until we get two numbers after the decimal point. If there is no decimal value, continue until the remainder is zero.

So, the square root of √518 is approximately 22.74.

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 518 using the approximation method.

Step 1: We have to find the closest perfect square of √518. The closest perfect square less than 518 is 484 (22²) and more than 518 is 529 (23²). Therefore, √518 falls between 22 and 23.

Step 2: Apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Using the formula: (518 - 484) / (529 - 484) = 34 / 45 ≈ 0.756

The approximate square root is the whole number plus the decimal: 22 + 0.756 ≈ 22.756.

• Square root: A square root is the inverse of a square. Example: 4² = 16, and the inverse of the square is the square root, which 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, but the positive square root is more commonly used due to its applications in the real world. This is known as the principal square root.

• Prime factorization: The process of expressing a number as a product of its prime factors.

• Decimal: A number that includes a whole number and a fractional part, represented with a decimal point, such as 22.74.