Square Root of 5680

The square root is the inverse of the square of a number. 5680 is not a perfect square. The square root of 5680 is expressed in both radical and exponential forms. In radical form, it is expressed as √5680, whereas in exponential form, it is expressed as (5680)^(1/2). √5680 ≈ 75.368, 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 used for perfect square numbers. However, for non-perfect square numbers, 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 5680 is broken down into its prime factors.

Step 1: Finding the prime factors of 5680 Breaking it down, we get 2 x 2 x 2 x 2 x 5 x 71: 2^4 x 5^1 x 71^1

Step 2: Now we found out the prime factors of 5680. The second step is to make pairs of those prime factors. Since 5680 is not a perfect square, the digits of the number can’t be grouped in pairs.

Therefore, calculating 5680 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 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 5680, we need to group it as 80 and 56.

Step 2: Now we need to find n whose square is ≤ 56. We can say n is ‘7’ because 7 x 7 is 49, which is less than or equal to 56. Now the quotient is 7, and after subtracting 49 from 56, the remainder is 7.

Step 3: Now let us bring down 80, which is the new dividend. Add the old divisor with the same number 7 + 7 to get 14, which will be our new divisor.

Step 4: The new divisor will be the sum of the dividend and quotient. Now we get 14n as the new divisor, and we need to find the value of n.

Step 5: The next step is finding 14n x n ≤ 780. Let us consider n as 5; now 145 x 5 = 725.

Step 6: Subtracting 725 from 780, the difference is 55, and the quotient is 75.

Step 7: Since the dividend is less than the divisor, we need to add a decimal point. Adding the decimal point allows us to add two zeroes to the dividend. Now the new dividend is 5500.

Step 8: Now we need to find the new divisor that is 150 because 1503 x 3 = 4509.

Step 9: Subtracting 4509 from 5500, we get the result 991.

Step 10: Now the quotient is 75.3.

Step 11: Continue doing these steps until we get two numbers after the decimal point. If there are no decimal values, continue until the remainder is zero.

So the square root of √5680 is approximately 75.368.

The approximation method is another method for finding square roots. It is an easy way to find the square root of a given number. Let us learn how to find the square root of 5680 using the approximation method.

Step 1: Now we have to find the closest perfect squares to √5680.

The smallest perfect square less than 5680 is 5625, and the largest perfect square more than 5680 is 5776. √5680 falls somewhere between 75 and 76.

Step 2: Now we need to apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Going by the formula, (5680 - 5625) ÷ (5776 - 5625) = 0.368

Using the formula, we identified the decimal point of our square root. The next step is adding the value we got initially to the decimal number, which is 75 + 0.368 = 75.368, so the square root of 5680 is approximately 75.368.

• Square root: A square root is the inverse of a square. Example: 4^2 = 16, and the inverse of the square is the square root, that is, √16 = 4.

• Irrational number: An irrational number is a number that cannot be written in the form of p/q, where q is not equal to zero and p and q are integers.

• Long division method: A method used to find the square root of non-perfect squares by breaking them into pairs and calculating step by step.

• Perfect square: A number that is the square of an integer. For example, 144 is a perfect square because it is 12^2.

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