Square Root of 521

The square root is the inverse of the square of the number. 521 is not a perfect square. The square root of 521 is expressed in both radical and exponential form. In the radical form, it is expressed as √521, whereas (521)^(1/2) in the exponential form. √521 ≈ 22.82542, 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 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 prime factorization of a number is the product of its prime factors. Now let us look at how 521 is broken down into its prime factors.

Step 1: Finding the prime factors of 521 521 is a prime number itself and cannot be broken down further into smaller prime factors.

Step 2: Since 521 is not a perfect square, calculating its square root 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 521, we group it as 21 and 5.

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 × 2 = 4, which is less than 5. Now the quotient is 2, and after subtracting 4 from 5, the remainder is 1.

Step 3: Now let us bring down 21, which is the new dividend. Add the old divisor with the same number 2 + 2 to get 4, which will be our new divisor.

Step 4: The new divisor will be 4n. We need to find the value of n.

Step 5: The next step is finding 4n × n ≤ 121. Let us consider n as 2, now 42 × 2 = 84.

Step 6: Subtract 84 from 121, the difference is 37, 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 3700.

Step 8: Now we need to find the new divisor, which is 229 because 229 × 9 = 2061.

Step 9: Subtracting 2061 from 3700, we get the result 1639.

Step 10: Now the quotient is 22.8.

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

So the approximate square root of √521 is 22.82.

The approximation method is another method for finding square roots, and 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 521 using the approximation method.

Step 1: Now we have to find the closest perfect squares surrounding √521. The smallest perfect square less than 521 is 484, and the largest perfect square greater than 521 is 529. √521 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 (521 - 484) ÷ (529 - 484) = 0.82 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 + 0.82 = 22.82, so the square root of 521 is approximately 22.82.

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

• Prime number: A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself.

• Decimal: A decimal is a number that has a whole number and a fraction in a single number, for example, 7.86, 8.65, and 9.42.

• Approximation method: The approximation method is a technique used to find an estimated value of the square root of a number, especially when it is not a perfect square.