Square Root of 468

The square root is the inverse of the square of a number. 468 is not a perfect square. The square root of 468 is expressed in both radical and exponential form. In the radical form, it is expressed as √468, whereas (468)^(1/2) in the exponential form. √468 ≈ 21.6333, 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:

• 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 468 is broken down into its prime factors:

Step 1: Finding the prime factors of 468

Breaking it down, we get 2 x 2 x 3 x 3 x 13: 2^2 x 3^2 x 13

Step 2: Now we found out the prime factors of 468. The second step is to make pairs of those prime factors. Since 468 is not a perfect square, the digits of the number can’t be grouped in pairs.

Therefore, calculating 468 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 468, we need to group it as 68 and 4.

Step 2: Now we need to find n whose square is 4. We can say n as ‘2’ because 2 x 2 is equal to 4. Now the quotient is 2 and the remainder is 0.

Step 3: Now let us bring down 68 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.

Step 5: The next step is finding 4n x n ≤ 68. Let us consider n as 1, now 41 x 1 = 41.

Step 6: Subtract 68 from 41, the difference is 27, and the quotient is 21.

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 zeros to the dividend. Now the new dividend is 2700.

Step 8: Now we need to find the new divisor. Let us consider n as 6, now 426 x 6 = 2556.

Step 9: Subtracting 2556 from 2700, we get the result 144.

Step 10: Now the quotient is 21.6

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

So the square root of √468 is approximately 21.63.

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 468 using the approximation method.

Step 1: Now we have to find the closest perfect square of √468.

The smallest perfect square less than 468 is 441 and the largest perfect square greater than 468 is 484. √468 falls somewhere between 21 and 22.

Step 2: Now we need to apply the formula: (Given number - smaller perfect square) ÷ (Larger perfect square - smaller perfect square).

Using the formula (468 - 441) ÷ (484 - 441) = 27 ÷ 43 ≈ 0.6279. 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 21 + 0.6279 ≈ 21.63, so the square root of 468 is approximately 21.63.

• 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 the principal square root.
   
• 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.
   
• Prime factorization: Expressing a number as the product of its prime factors is called prime factorization. For example, the prime factorization of 468 is 2 x 2 x 3 x 3 x 13.