Square Root of 452

The square root is the inverse of the square of the number. 452 is not a perfect square. The square root of 452 is expressed in both radical and exponential form. In radical form, it is expressed as √452, whereas \(452^{1/2}\) in exponential form. √452 ≈ 21.2603, 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 452 is broken down into its prime factors:

Step 1: Finding the prime factors of 452 Breaking it down, we get 2 x 2 x 113: (2^2 times 113)

Step 2: Now we found the prime factors of 452. Since 452 is not a perfect square, the digits of the number can’t be grouped in pairs.

Therefore, calculating 452 using prime factorization is not straightforward.

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 452, we need to group it as 52 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 times 2 = 4). Now the quotient is 2. After subtracting 4 from 4, the remainder is 0.

Step 3: Now, bring down 52, 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 times n leq 52). Let us consider n as 1, now (41 times 1 = 41).

Step 6: Subtract 52 from 41; the difference is 11, 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 1100.

Step 8: Now we need to find the new divisor by considering n as 2 because (422 times 2 = 844).

Step 9: Subtracting 844 from 1100 gives the result 256.

Step 10: Now the quotient is 21.2.

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

So, the square root of √452 is approximately 21.26.

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 452 using the approximation method.

Step 1: Now we have to find the closest perfect squares around √452. The smallest perfect square less than 452 is 441, and the largest perfect square greater than 452 is 484. √452 falls somewhere between 21 and 22.

Step 2: Now we need to apply the formula: Using the formula ((452 - 441) div (484 - 441) = 11/43 approx 0.256).

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

The next step is adding the value we got initially to the decimal number, which is (21 + 0.256 = 21.256), so the square root of 452 is approximately 21.26.

• 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.

• Perfect square: A perfect square is a number that can be expressed as the square of an integer. Example: 16 is a perfect square because it is 4².

• Decimal: If a number has a whole number and a fraction in a single number, then it is called a decimal. Example: 7.86, 8.65, and 9.42 are decimals.

• Approximation: The process of finding a value that is close enough to the right answer, usually with a degree of accuracy specified.