Square Root of 7056

The square root is the inverse of the square of the number. 7056 is a perfect square. The square root of 7056 is expressed in both radical and exponential form. In radical form, it is expressed as √7056, whereas in exponential form it is expressed as (7056)¹/². √7056 = 84, 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. For non-perfect square numbers, the long-division method and approximation method are commonly used. Let's explore 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 7056 is broken down into its prime factors:

Step 1: Finding the prime factors of 7056 Breaking it down, we get 2 x 2 x 2 x 2 x 3 x 3 x 7 x 7: 2⁴ x 3² x 7²

Step 2: Now we have found the prime factors of 7056. The second step is to make pairs of those prime factors. Since 7056 is a perfect square, we can make pairs of the prime factors: (2 x 2) x (2 x 2) x (3 x 3) x (7 x 7)

Step 3: Taking one number from each pair gives us 2 x 2 x 3 x 7 = 84.

Therefore, the square root of 7056 is 84.

The long division method is used for both perfect and non-perfect square numbers. Here, let's 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 pairs. In the case of 7056, we need to group it as 56 and 70.

Step 2: Find the largest number whose square is less than or equal to 70. That number is 8 because 8 x 8 = 64 is less than 70. Now, the quotient is 8, and the remainder is 70 - 64 = 6.

Step 3: Bring down the next pair, which is 56, making the new dividend 656. Double the quotient (8) to get 16, which is used as the first part of the new divisor.

Step 4: Find a number n such that 16n x n is less than or equal to 656. The suitable value for n is 4 because 164 x 4 = 656.

Step 5: Since there is no remainder left, the quotient 84 is the square root of 7056.

The approximation method is another way to find the square roots, especially for non-perfect squares. However, since 7056 is a perfect square, we can directly find its square root.

Step 1: Identify the closest perfect squares around 7056. Since 7056 is a perfect square itself, we find that √7056 = 84 directly.

Students often make mistakes while finding the square root, such as forgetting about the negative square root or skipping steps in the long division method. Let's explore some common mistakes in detail:

• Square root: A square root is a value that, when multiplied by itself, gives the original number. Example: The square root of 16 is 4, i.e., √16 = 4.
   
• Perfect square: A perfect square is a number that can be expressed as the product of an integer with itself. Example: 64 is a perfect square because 8 x 8 = 64.
   
• Rational number: A rational number is a number that can be expressed as the quotient or fraction of two integers, where the denominator is not zero.
   
• Prime factorization: Prime factorization is the process of expressing a number as the product of its prime numbers.
   
• Long division method: The long division method is a technique for finding the square root of a number by dividing it into smaller parts.