Square Root of 421

The square root is the inverse of the square of the number. 421 is not a perfect square. The square root of 421 is expressed in both radical and exponential form. In the radical form, it is expressed as √421, whereas (421)^(1/2) in the exponential form. √421 ≈ 20.5183, 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:

• Prime factorization method

• Long division method

• Approximation method

The product of prime factors is the prime factorization of a number. Now let us look at how 421 is broken down into its prime factors.

Step 1: Finding the prime factors of 421 421 is already a prime number. Therefore, it cannot be broken down further.

Step 2: Since 421 is not a perfect square, calculating √421 using prime factorization directly 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 421, we can group it as 21 and 4.

Step 2: Now we need to find n whose square is ≤ 4. We can say n is 2 because 2 × 2 = 4. Now the quotient is 2, and after subtracting 4 from 4, the remainder is 0.

Step 3: Now let us bring down 21, which is the new dividend. Add the old divisor with the same number, 2 + 2 = 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 ≤ 21. Let us consider n as 5, now 4 × 5 = 20, and 20 × 5 = 100, which is too large. Adjust n accordingly.

Step 6: Subtract 21 from 20, the difference is 1, and the quotient is 20.

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 100.

Step 8: Now we need to find the new divisor that is approximately 204, as 204 × 5 = 1020.

Step 9: Adjust and continue the process until sufficient decimal accuracy is achieved.

Step 10: The quotient becomes approximately 20.518.

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 421 using the approximation method.

Step 1: Now we have to find the closest perfect square of √421. The smallest perfect square before 421 is 400, and the largest perfect square after 421 is 441. √421 falls somewhere between 20 and 21.

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 (421 - 400) / (441 - 400) ≈ 0.512. 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 20 + 0.518 ≈ 20.518, so the square root of 421 is approximately 20.518.

• 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 natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers.

• Approximation: An approximation is a value or quantity that is nearly but not exactly correct, used particularly in mathematical contexts to simplify complex calculations.

• Long division method: A mathematical method of dividing large numbers by breaking down the division into a series of easier steps. This method is often used to find square roots of non-perfect squares.