Square Root of 5025

The square root is the inverse operation of squaring a number. 5025 is not a perfect square. The square root of 5025 can be expressed in both radical and exponential forms. In the radical form, it is expressed as √5025, whereas in exponential form it is expressed as (5025)^(1/2). √5025 ≈ 70.882, 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 typically used for perfect square numbers. However, for non-perfect square numbers, methods such as long division and approximation 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's look at how 5025 breaks down into its prime factors:

Step 1: Finding the prime factors of 5025

Breaking it down, we get 3 x 3 x 5 x 5 x 67: 3^2 x 5^2 x 67

Step 2: The prime factors of 5025 include pairs of 3 and 5. Since 5025 is not a perfect square, we cannot group all prime factors into pairs. Therefore, calculating √5025 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's learn how to find the square root using the long division method, step by step:

Step 1: Start by grouping the numbers from right to left. For 5025, group it as 50 and 25.

Step 2: Find n whose square is less than or equal to 50. Here, n is 7 because 7 x 7 = 49, which is less than 50. Subtract 49 from 50 to get a remainder of 1, and bring down 25, making the new dividend 125.

Step 3: Double the quotient (7) to get 14, which becomes part of our new divisor.

Step 4: Find a digit x such that 14x times x is less than or equal to 125. Here, x is 8 because 148 times 8 = 1184, which is less than 1250. Subtract 1184 from 1250 to get a remainder of 66.

Step 5: Continue the process by bringing down pairs of zeros and repeating the steps to find the next digits. Eventually, you'll approximate √5025 to 70.882.

The approximation method is another way to find square roots. It is an easy method to estimate the square root of a given number. Here's how to find the square root of 5025 using the approximation method:

Step 1: Identify the closest perfect squares around 5025. The smallest perfect square less than 5025 is 4900 (70^2), and the largest perfect square greater than 5025 is 5184 (72^2). So, √5025 falls between 70 and 72.

Step 2: Apply linear interpolation: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square) (5025 - 4900) / (5184 - 4900) = 125 / 284 Using this, √5025 is approximately 70.882.

• Square root: The square root of a number is a value that, when multiplied by itself, gives the original number. Example: The square root of 16 is 4 because 4 x 4 = 16.

• Irrational number: An irrational number cannot be expressed as a simple fraction. It has a non-repeating, non-terminating decimal expansion.

• Principal square root: The principal square root is the positive square root of a number. It is the most commonly used in practical applications.

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

• Decimal: A decimal is a number that consists of a whole number and a fractional part separated by a decimal point, such as 7.86 or 8.65.