Square Root of 523

The square root is the inverse of the square of the number. 523 is not a perfect square. The square root of 523 is expressed in both radical and exponential form. In the radical form, it is expressed as √523, whereas (523)^(1/2) in the exponential form. √523 ≈ 22.84732, 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 523 is broken down into its prime factors.

Step 1: Finding the prime factors of 523

523 is a prime number itself, so it cannot be broken down into smaller prime factors.

Since 523 is not a perfect square, therefore, calculating its square root 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 523, it remains as 523.

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

Step 3: Now let us bring down 23, making the new dividend 123. Add the old divisor (2) with itself to get 4, which will be our new divisor.

Step 4: Now, we need to find a number n such that 4n x n is less than or equal to 123. Let n be 2, then 42 x 2 = 84.

Step 5: Subtract 84 from 123, the remainder is 39, and the partial 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 3900.

Step 7: Now we need to find the new divisor. Using trial and error, we find the new divisor to be 445 because 445 x 9 = 4005, which is too large. Using 444 x 8 = 3552, which is closer.

Step 8: Subtract 3552 from 3900 to get 348.

Step 9: Now the quotient is 22.8. Continue doing these steps until you get two numbers after the decimal point or until the remainder is zero.

So, the square root of √523 ≈ 22.85.

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 523 using the approximation method.

Step 1: Find the closest perfect squares of √523. The smallest perfect square less than 523 is 484 (22²) and the largest perfect square greater than 523 is 529 (23²). √523 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).

Using the formula (523 - 484) ÷ (529 - 484) = 39 ÷ 45 = 0.8667. 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 + 0.87 ≈ 22.87. So, the square root of 523 is approximately 22.87.

• 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.

• Prime number: A prime number is a number that has exactly two distinct positive divisors: 1 and itself.

• Long division method: A method used to find the square root of a non-perfect square by dividing the number into groups and estimating the root iteratively.

• Approximation: A method of finding an approximate value of a square root by identifying the nearest perfect squares and estimating the decimal value in between.