Square Root of 514

The square root is the inverse of the square of the number. 514 is not a perfect square. The square root of 514 is expressed in both radical and exponential form. In the radical form, it is expressed as √514, whereas (514)^(1/2) in the exponential form. √514 ≈ 22.67157, 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, 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 514 is broken down into its prime factors.

Step 1: Finding the prime factors of 514 Breaking it down, we get 2 x 257.

Step 2: Now we found the prime factors of 514.

Since 514 is not a perfect square, calculating its square root using prime factorization isn't straightforward.

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 514, we need to group it as 14 and 5.

Step 2: Now we need to find n whose square is closest to 5 without exceeding it. We can say n as ‘2’ because 2 x 2 = 4, which is less than or equal to 5. Now the quotient is 2. After subtracting 4 from 5, the remainder is 1.

Step 3: Bring down 14, making the new dividend 114. Add the old divisor with the same number 2 + 2 = 4, which will be our new divisor.

Step 4: Find the value of n such that 4n x n ≤ 114. Let n = 2, so 42 x 2 = 84.

Step 5: Subtract 84 from 114, the difference is 30, 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 3000.

Step 7: The next step is finding the new divisor. Try n = 7, as 447 x 7 = 3129, which exceeds 3000. Try n = 6, as 446 x 6 = 2676.

Step 8: Subtracting 2676 from 3000 gives 324.

Step 9: The quotient is approximately 22.67. Continue the process to obtain more decimal places if necessary.

So the square root of √514 is approximately 22.67157.

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 514 using the approximation method.

Step 1: Now we have to find the closest perfect squares around √514. The smallest perfect square less than 514 is 484 (22²), and the largest perfect square greater than 514 is 529 (23²). √514 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 (514 - 484) / (529 - 484) = 30 / 45 = 0.6667.

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 + 0.6667 ≈ 22.67.

Therefore, the square root of 514 is approximately 22.67157.

• Square root: A square root is the inverse of a square. For 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.

• Long Division Method: A step-by-step method used to find square roots of non-perfect squares by dividing the number into smaller parts.

• Approximation Method: A technique used to estimate the square root of a number by finding the nearest perfect squares and using a proportional relationship.

• Decimal: A number that includes a whole number and a fractional part separated by a decimal point, such as 22.67.