Square Root of 7569

The square root is the inverse of the square of the number. 7569 is a perfect square. The square root of 7569 is expressed in both radical and exponential form. In the radical form, it is expressed as √7569, whereas (7569)^(1/2) in exponential form. √7569 = 87, 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. However, for non-perfect square numbers, methods like 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 7569 is broken down into its prime factors.

Step 1: Finding the prime factors of 7569 Breaking it down, we get 3 x 3 x 29 x 29: 3² x 29²

Step 2: Now we found out the prime factors of 7569. The second step is to make pairs of those prime factors. Since 7569 is a perfect square, we can pair the prime factors. Therefore, calculating √7569 using prime factorization yields 87 (since 3 x 29 = 87).

The long division method can also be used for finding the square root of a perfect square. Let us now learn how to find the square root 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 7569, we group it as 75 and 69.

Step 2: Now we need to find n whose square is less than or equal to 75. We can say n is '8' because 8 x 8 = 64, which is less than 75. Now the quotient is 8, and after subtracting 75 - 64, the remainder is 11.

Step 3: Now let us bring down 69, which is the new dividend. Add the old divisor with the same number 8 + 8, we get 16, which will be our new divisor.

Step 4: The new divisor will be 16n. We need to find the value of n.

Step 5: The next step is finding 16n × n ≤ 1169. Let us consider n as 7, now 167 × 7 = 1169.

Step 6: Subtract 1169 from 1169, and the remainder is 0, and the quotient is 87. So the square root of √7569 is 87.

Approximation method is generally used for non-perfect squares, but here's how we can confirm the square root of 7569 using this method.

Step 1: Now, we have to find the closest perfect squares around √7569. The smallest perfect square close to 7569 is 7225, and the largest perfect square is 7744. √7569 falls somewhere between 85 and 88.

Step 2: Now, we need to apply the approximation method, which involves checking values in this range. Since 87² = 7569, we confirm that 87 is the square root of 7569.

• Square root: A square root is the inverse of a square. Example: 9² = 81, and the inverse of the square is the square root, that is √81 = 9.
   
• 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 perfect square is a number that is the square of an integer. For example, 7569 is a perfect square because it is 87².
   
• Prime factorization: Prime factorization is expressing a number as the product of its prime factors.?
   
• Long division method: A method used to find the square root of a number by dividing and averaging.