Square Root of 485

The square root is the inverse of the square of the number. 485 is not a perfect square. The square root of 485 is expressed in both radical and exponential forms.

In the radical form, it is expressed as √485, whereas (485)^(1/2) in the exponential form. √485 ≈ 22.02272, 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 485 is broken down into its prime factors.

Step 1: Finding the prime factors of 485

Breaking it down, we get 5 x 97: 5¹ x 97¹

Step 2: Now we found out the prime factors of 485. The second step is to make pairs of those prime factors. Since 485 is not a perfect square, the digits of the number can’t be grouped in pairs.

Therefore, calculating 485 using prime factorization is impossible.

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 485, we need to group it as 85 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 x 2 is 4. Now the quotient is 2, and after subtracting 4 - 4, the remainder is 0.

Step 3: Now let us bring down 85, 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 4n. We need to find the value of n.

Step 5: The next step is finding 4n x n ≤ 85. Let us consider n as 2, now 42 x 2 = 84.

Step 6: Subtract 85 from 84, the difference is 1, 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 100.

Step 8: Now we need to find the new divisor that is 441 because 441 x 1 = 441.

Step 9: Subtracting 441 from 100 we get the result 59.

Step 10: Now the quotient is 22.0.

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

So the square root of √485 is approximately 22.02.

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 485 using the approximation method.

Step 1: Now we have to find the closest perfect square of √485. The smallest perfect square less than 485 is 484, and the largest perfect square greater than 485 is 529. √485 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). Going by the formula (485 - 484) / (529 - 484) = 0.02222

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 + 0.02222 ≈ 22.02, so the square root of 485 is approximately 22.02.

• 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.
   
• Principal square root: A number has both positive and negative square roots. However, it is always a positive square root that has more prominence due to its uses in the real world. That is the reason it is also known as the principal square root.
   
• Prime factorization: It is the expression of a number as the product of its prime factors. For example, the prime factorization of 485 is 5 x 97.
   
• 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.