Square Root of 7225

The square root is the inverse of the square of a number. 7225 is a perfect square. The square root of 7225 is expressed in both radical and exponential form. In the radical form, it is expressed as √7225, whereas (7225)^(1/2) in the exponential form. √7225 = 85, 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 can be used for perfect square numbers. The long division method and approximation method can also be useful. 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 7225 is broken down into its prime factors:

Step 1: Finding the prime factors of 7225 Breaking it down, we get 5 × 5 × 17 × 17. This can be expressed as (5^2) × (17^2).

Step 2: Now we found out the prime factors of 7225. The second step is to take the square root of the prime factors, which results in 5 × 17 = 85. Therefore, the square root of 7225 is 85.

The long division method is particularly useful for larger numbers. Let us now learn how to find the square root using the long division method, step by step

Step 1: Pair the digits of 7225 from right to left, which gives us the pairs (72)(25).

Step 2: Find the largest number whose square is less than or equal to 72. That number is 8 because 8 × 8 = 64. Now, the quotient is 8, and the remainder is 72 - 64 = 8.

Step 3: Bring down the next pair 25 to the right of the remainder. The new dividend is 825.

Step 4: Double the current quotient, 8, to get 16. This is the starting digit of the new divisor.

Step 5: Determine the largest digit (n) such that (160 + n) × n is less than or equal to 825. Here, n is 5 because 165 × 5 = 825.

Step 6: Subtract 825 from 825, resulting in a remainder of 0. The quotient thus obtained is 85, which is the square root of 7225.

The approximation method involves finding the square roots of numbers near the given number. Here is how it's done for 7225:

Step 1: Identify the closest perfect squares around 7225. The perfect squares are 7056 (84^2) and 7225 (85^2).

Step 2: Since 7225 itself is a perfect square, there is no need for further approximation. The square root of 7225 is exactly 85.

• Square root: A square root is the inverse of a square. For example, 85^2 = 7225, and the inverse of the square is the square root, so √7225 = 85.
   
• Rational number: A rational number is a number that can be written in the form of p/q, where q is not equal to zero and p and q are integers.
   
• Perfect square: A number whose square root is an integer. For example, 7225 is a perfect square because its square root is 85, an integer.
   
• Prime factorization: The expression of a number as the product of its prime factors. For example, the prime factorization of 7225 is 5 × 5 × 17 × 17.
   
• Long division method: A method used to find the square root of a number by dividing it into pairs of digits, finding a divisor, and iteratively solving for the quotient.