Square Root of 5050

The square root is the inverse of the square of a number. 5050 is not a perfect square. The square root of 5050 is expressed in both radical and exponential form. In radical form, it is expressed as √5050, whereas (5050)^(1/2) in exponential form. √5050 ≈ 71.0633, 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 5050 is broken down into its prime factors.

Step 1: Finding the prime factors of 5050.

Breaking it down, we get 2 x 5 x 5 x 101: 2^1 x 5^2 x 101^1

Step 2: Now we have found the prime factors of 5050. The second step is to make pairs of those prime factors. Since 5050 is not a perfect square, the digits of the number can’t be grouped in pairs. Therefore, calculating 5050 using prime factorization for a perfect square is not feasible.

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: Group the digits from right to left. In the case of 5050, group it as 50 and 50.

Step 2: Find n whose square is less than or equal to 50. Here, n is 7 because 7 x 7 = 49. Now the quotient is 7, and after subtracting 49 from 50, the remainder is 1.

Step 3: Bring down the next pair of digits, 50, making it 150. Add the old divisor with the same number: 7 + 7 = 14, which will be our new divisor.

Step 4: Find n such that 14n x n ≤ 150. Let's consider n as 1, now 141 x 1 = 141.

Step 5: Subtract 141 from 150, the difference is 9, and the quotient is 71.

Step 6: 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 900.

Step 7: Find the new divisor, which is 711 because 711 x 1 = 711.

Step 8: Subtract 711 from 900, resulting in 189.

Step 9: Now the quotient is 71.1.

Step 10: Continue these steps until we have sufficient decimal places or the remainder is zero.

So the square root of √5050 ≈ 71.0633.

The approximation method is another method for finding square roots, which is an easy method to find the square root of a given number. Now let us learn how to find the square root of 5050 using the approximation method.

Step 1: Find the closest perfect squares around √5050. The smallest perfect square less than 5050 is 4900 (70^2), and the largest perfect square greater than 5050 is 5184 (72^2). √5050 falls somewhere between 70 and 72.

Step 2: Apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Using the formula (5050 - 4900) ÷ (5184 - 4900) = 150 ÷ 284 ≈ 0.528. Adding the value to the closest lower square root gives 70 + 0.528 ≈ 70.528. Plain approximation yields a value of approximately 71.0633, so the square root of 5050 is approximately 71.0633.

• Square root: A square root is the inverse of a square. Example: 4^2 = 16, and the square root is √16 = 4.

• Irrational number: An irrational number is a number that cannot be written in the form p/q, where q is not zero, and p and q are integers.

• Principal square root: A number has both positive and negative square roots, but the positive square root is more prominent in real-world applications. This is known as the principal square root.

• Prime factorization: The process of expressing a number as the product of its prime factors.

• Decimal: A number that includes a whole number and a fractional part, such as 7.86, 8.65, or 9.42.