Square Root of 403

The square root is the inverse of the square of the number. 403 is not a perfect square. The square root of 403 is expressed in both radical and exponential form. In the radical form, it is expressed as √403, whereas (403)^(1/2) in the exponential form. √403 ≈ 20.07486, 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 403 is broken down into its prime factors:

Step 1: Finding the prime factors of 403 Breaking it down, we get 13 × 31

Step 2: Now we found out the prime factors of 403. The second step is to make pairs of those prime factors. Since 403 is not a perfect square, the digits of the number can’t be grouped in pairs.

Therefore, calculating 403 using prime factorization alone 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 403, we need to group it as 03 and 4.

Step 2: Now we need to find n whose square is less than or equal to 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 03, 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 such that 4n × n ≤ 3. This is not possible, so we add a decimal point after the quotient and bring down two zeroes.

Step 5: The new dividend is 300. Find n such that 40n × n ≤ 300. Taking n as 7, we find 407 × 7 = 2849.

Step 6: Subtract 2849 from 3000, the difference is 151, and the quotient becomes 20.7.

Step 7: Since the dividend is less than the divisor, we need to add more zeroes. The new dividend becomes 15100.

Step 8: Now find the new divisor, which is 407. The next digit is 3 because 4073 × 3 = 12219.

Step 9: Subtract 12219 from 15100 to get 2881. The quotient becomes 20.73.

Step 10: Continue doing these steps until we get the desired precision.

The square root of √403 is approximately 20.07486.

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 403 using the approximation method.

Step 1: Now we have to find the closest perfect squares around √403.

The smallest perfect square less than 403 is 400, and the largest perfect square greater than 403 is 441.

√403 falls somewhere between 20 and 21.

Step 2: Now we need to apply the formula:

(Given number - smallest perfect square) / (Greater perfect square - smallest perfect square).

Going by the formula (403 - 400) ÷ (441 - 400) = 3 ÷ 41 ≈ 0.073.

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 + 0.073 = 20.073, so the square root of 403 is approximately 20.074.

• Square root: A square root is the inverse of a square. Example: 4^2 = 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 a principal square root
   
• Prime factorization: The representation of a number as the product of its prime numbers. For example, 403 = 13 × 31.
   
• 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.