Square Root of 450

The square root is the inverse of the square of a number. 450 is not a perfect square. The square root of 450 is expressed in both radical and exponential form. In the radical form, it is expressed as √450, whereas (450)^(1/2) in the exponential form. √450 ≈ 21.2132, 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 450 is broken down into its prime factors.

Step 1: Finding the prime factors of 450 Breaking it down, we get 2 x 3 x 3 x 5 x 5: 2^1 x 3^2 x 5^2

Step 2: Now we found out the prime factors of 450.

The second step is to make pairs of those prime factors. Since 450 is not a perfect square, the digits of the number can’t be grouped in complete pairs.

Therefore, calculating √450 using prime factorization involves finding √(3 x 5 x 5 x 2 x 3) = 3 x 5 x √2 x √3 = 15√6.

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 450, we need to group it as 50 and 4.

Step 2: Now we need to find n whose square is ≤ 4. We can say n as ‘2’ because 2 x 2 = 4 is lesser than or equal to 4. Now the quotient is 2, after subtracting 4 - 4, the remainder is 0.

Step 3: Now let us bring down 50, 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 × n ≤ 50. Let us consider n as 1, now 41 x 1 = 41.

Step 6: Subtract 50 from 41; the difference is 9, 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 zeroes to the dividend. Now the new dividend is 900.

Step 8: Now we need to find the new divisor that is 43 because 432 x 2 = 864.

Step 9: Subtracting 864 from 900, we get the result 36.

Step 10: Now the quotient is 21.2.

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

So the square root of √450 is approximately 21.21.

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 450 using the approximation method.

Step 1: Now we have to find the closest perfect square of √450. The smallest perfect square less than 450 is 441, and the largest perfect square greater than 450 is 484. √450 falls somewhere between 21 and 22.

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 (450 - 441) ÷ (484-441) = 9/43 ≈ 0.209.

Using the formula, we identified the decimal point of our square root.

The next step is adding the initial whole number to the decimal number, which is 21 + 0.209 ≈ 21.21, so the square root of 450 is approximately 21.21.

• 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.

• Perfect square: A perfect square is a number that is the square of an integer. Example: 9 is a perfect square because it is 3^2.

• Prime factorization: Prime factorization is expressing a number as a product of its prime factors. Example: 450 = 2 x 3^2 x 5^2.

• 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.