Square Root of 517

The square root is the inverse of the square of the number. 517 is not a perfect square. The square root of 517 is expressed in both radical and exponential form. In the radical form, it is expressed as √517, whereas (517)^(1/2) in the exponential form. √517 ≈ 22.722, 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 517 is broken down into its prime factors.

Step 1: Finding the prime factors of 517 Breaking it down, we get 11 × 47: 11¹ × 47¹

Step 2: Now we found the prime factors of 517. Since 517 is not a perfect square, the digits of the number can’t be grouped in pairs.

Therefore, calculating 517 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 517, we need to group it as 17 and 5.

Step 2: Now we need to find n whose square is 5. We can say n as ‘2’ because 2 × 2 is lesser than or equal to 5. Now the quotient is 2, and after subtracting 4 from 5, the remainder is 1.

Step 3: Now let us bring down 17, which is the new dividend. Add the old divisor with the same number 2 + 2, we get 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 × n ≤ 117. Let us consider n as 2, now 42 × 2 = 84.

Step 6: Subtract 84 from 117; the difference is 33, 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 3300.

Step 8: Now we need to find the new divisor, which is 449, because 449 × 7 = 3143.

Step 9: Subtracting 3143 from 3300, we get the result 157.

Step 10: Now the quotient is 22.7.

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

So the square root of √517 is approximately 22.72.

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 517 using the approximation method.

Step 1: Now we have to find the closest perfect square of √517. The smallest perfect square less than 517 is 484, and the largest perfect square greater than 517 is 529. √517 falls somewhere between 22 and 23.

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 (517 - 484) ÷ (529 - 484) = 33 ÷ 45 ≈ 0.733

.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 + 0.733 = 22.733, so the square root of 517 is approximately 22.733.

• 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 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 process of breaking down a number into its basic prime factors. For example, the prime factorization of 517 is 11 × 47.

• 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.