Square Root of 7025

The square root is the inverse operation of squaring a number. Since 7025 is not a perfect square, its square root is expressed in both radical and exponential forms. In radical form, it is expressed as √7025, whereas in exponential form, it is (7025)^(1/2). The approximate value of √7025 is 83.8109, which is an irrational number because it cannot be exactly expressed as a fraction p/q, where p and q are integers and q ≠ 0.

For perfect square numbers, the prime factorization method is useful. However, for non-perfect square numbers like 7025, methods such as the long division and approximation methods are used. Let us learn these methods:

• Prime factorization method
• Long division method
• Approximation method

The product of prime factors is known as the prime factorization of a number. Let's examine how 7025 is decomposed into its prime factors:

Step 1: Identify the prime factors of 7025. Breaking it down, we get 5 x 5 x 281: 5^2 x 281^1

Step 2: Now that we have identified the prime factors of 7025, the next step is to pair those prime factors. Since 7025 is not a perfect square, the digits cannot be grouped into perfect pairs, making it impossible to calculate √7025 using prime factorization alone.

The long division method is particularly used for non-perfect squares. In this method, we find the closest perfect square number to the given number. Let us learn how to find the square root using the long division method, step by step:

Step 1: Group the numbers from right to left. For 7025, group as 25 and 70.

Step 2: Find a number whose square is less than or equal to 70. This number is 8 because 8^2 = 64. The quotient is 8 and the remainder is 70 - 64 = 6.

Step 3: Bring down 25, making the new dividend 625. Add 8 (the previous divisor) to itself to get 16, which is our new divisor base.

Step 4: Find a digit n such that 16n × n ≤ 625. Let n be 3, then 163 × 3 = 489.

Step 5: Subtract 489 from 625, giving a remainder of 136.

Step 6: Add decimal point and zeros to continue the division. The quotient so far is 83, and the process can continue to find more decimal places.

Step 7: Continue these steps until the required precision is achieved. So the square root of 7025 is approximately 83.8109.

The approximation method is another technique for finding square roots, providing an easy way to find the square root of a given number. Let's find the square root of 7025 using this method.

Step 1: Identify the closest perfect squares to 7025. The smallest perfect square is 6889 (83^2) and the largest is 7056 (84^2). √7025 falls between 83 and 84.

Step 2: Apply the formula: (Given number - smaller perfect square) / (Larger perfect square - smaller perfect square). Using the formula: (7025 - 6889) / (7056 - 6889) = 0.8109 Thus, the square root of 7025 is approximately 83 + 0.8109 = 83.8109.

• Square root: A square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 16 is 4 because 4 × 4 = 16.
   
• Irrational number: An irrational number cannot be expressed as a simple fraction. It has an infinite and non-repeating decimal expansion.
   
• Principal square root: The principal square root is the non-negative square root of a number.
   
• Decimal: A number that contains a decimal point, representing a whole number and fractional part.
   
• Prime factorization: Breaking down a number into its basic building blocks, which are prime numbers. For example, the prime factorization of 7025 is 5 × 5 × 281.