Square Root of 484

The square root is the inverse of the square of the number. 484 is a perfect square. The square root of 484 is expressed in both radical and exponential form.

In the radical form, it is expressed as √484, whereas (484)^(1/2) in the exponential form. √484 = 22, which is a rational number because it can 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 like 484. For non-perfect squares, methods like long division and approximation are used. Let us now learn how to find the square root using different methods: 

• Prime factorization method 
• Long division method 
• Verification through approximation

The prime factorization of a number involves expressing it as a product of its prime factors. Let us look at how 484 is broken down into its prime factors:

Step 1: Finding the prime factors of 484

Breaking it down, we get 2 x 2 x 11 x 11: 2² x 11²

Step 2: Now that we have found the prime factors of 484, the next step is to make pairs of those prime factors. Since 484 is a perfect square, its prime factors can be paired.

Step 3: The square root of 484 is the product of one number from each pair: 2 x 11 = 22.

The long division method is useful for finding the square root of both perfect and non-perfect square numbers. Let us learn how to find the square root of 484 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 484, we group it as 4 and 84.

Step 2: Now, we find n whose square is less than or equal to the first group, which is 4. Here, n is 2 because 2 x 2 = 4. The quotient is 2, and the remainder is 0.

Step 3: Bring down the next group, which is 84, making it the new dividend. Add the previous divisor (2) to itself to get a new divisor: 2 + 2 = 4.

Step 4: We find a number n such that 4n x n is less than or equal to 84. Here, n is 2 because 42 x 2 = 84.

Step 5: Subtract 84 from 84, resulting in a remainder of 0. The quotient is 22.

Step 6: Since there is no remainder, the process concludes here.

The square root of 484 is 22.

The approximation method can verify the square root of a perfect square like 484, though it is exact due to the nature of perfect squares.

Step 1: Identify the closest perfect squares around 484. These are 441 (21²) and 529 (23²).

Step 2: Since 484 is exactly between 441 and 529, it confirms that the square root is exactly 22.

Step 3: Therefore, the square root of 484 is precisely 22, confirming the results of the other methods.

• Square root: A square root is the inverse of a square. Example: 5² = 25, and the inverse of the square is the square root, so √25 = 5.
   
• Rational number: A rational number can be expressed as a fraction where the numerator and denominator are integers and the denominator is not zero.
   
• Perfect square: A perfect square is a number that is the square of an integer. For example, 144 is a perfect square because it is 12².
   
• Prime factorization: Prime factorization involves expressing a number as the product of its prime factors. For example, the prime factorization of 18 is 2 x 3².
   
• Perimeter: The perimeter is the total length around a two-dimensional shape. For a rectangle, it is calculated as 2 × (length + width).