Square Root of 720

The square root is the inverse of the square of the number. 720 is not a perfect square. The square root of 720 is expressed in both radical and exponential form. In the radical form, it is expressed as √720, whereas (720)^(1/2) in the exponential form. √720 ≈ 26.8328, 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, the prime factorization method is not used for non-perfect square numbers where 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 720 is broken down into its prime factors.

Step 1: Finding the prime factors of 720 Breaking it down, we get 2 x 2 x 2 x 2 x 3 x 3 x 5, which is 2^4 x 3^2 x 5.

Step 2: Now we found out the prime factors of 720. The second step is to make pairs of those prime factors. Since 720 is not a perfect square, there will be a single 5 left unpaired. Therefore, calculating √720 using prime factorization directly 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 720, we need to group it as 20 and 7.

Step 2: Now we need to find n whose square is less than or equal to 7. We can say n as '2' because 2 x 2 = 4, which is less than 7. Now the quotient is 2, after subtracting 4 from 7 the remainder is 3.

Step 3: Bring down the next pair, which is 20, making the new dividend 320.

Step 4: Double the current quotient (2), which gets us 4, our new partial divisor.

Step 5: Now we find n such that 4n x n ≤ 320. The correct n is 8, because 48 x 8 = 384, which is more than 320, while 47 x 7 = 329. Thus, n = 7, so 47 x 7 = 329.

Step 6: Subtract 320 from 329, the remainder is 9.

Step 7: Since the remainder 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, making it 900.

Step 8: Now we need to find the new divisor. Doubling the current quotient (27), we get 54.

Step 9: Find n such that 54n x n ≤ 900. The correct n is 1, because 541 x 1 = 541.

Step 10: Subtract 541 from 900, and the remainder is 359.

Step 11: Continue doing these steps until we get two numbers after the decimal point or the remainder is zero.

So the square root of √720 ≈ 26.83.

The approximation method is another method for finding square roots; it is an easy method to find the square root of a given number. Now let us learn how to find the square root of 720 using the approximation method.

Step 1: Now we have to find the closest perfect square of √720. The smallest perfect square less than 720 is 676 (26^2), and the largest perfect square greater than 720 is 729 (27^2). √720 falls somewhere between 26 and 27.

Step 2: Now we need to apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square) Using the formula (720 - 676) / (729 - 676) = 44 / 53 ≈ 0.83 Adding the approximate decimal to 26, we get 26 + 0.83 = 26.83.

So the approximate square root of 720 is 26.83.

• Square root: A square root is the inverse of squaring a number. For 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.

• Principal square root: A number has both positive and negative square roots; however, in practical scenarios, the positive square root is often used, known as the principal square root.

• Perfect square: A perfect square is a number that is the square of an integer. For example, 16 is a perfect square because it is 4 squared.

• Prime factorization: The process of expressing a number as the product of its prime factors. For example, the prime factorization of 720 is 2^4 x 3^2 x 5.