Square Root of 726

The square root is the inverse of squaring a number. Since 726 is not a perfect square, its square root is expressed in both radical and exponential forms. In radical form, it is expressed as √726, whereas in exponential form, it is expressed as (726)^(1/2). √726 ≈ 26.933, which is an irrational number because it cannot be expressed as a ratio of two integers.

The prime factorization method is applicable for perfect square numbers. For non-perfect squares like 726, the long division and approximation methods are often used. Let's explore these methods: 

• Prime factorization method 
• Long division method 
• Approximation method

The prime factorization of a number involves expressing it as a product of prime numbers. Let's explore the prime factorization of 726.

Step 1: Finding the prime factors of 726

Breaking it down, we get 2 x 3 x 11 x 11: 2^1 x 3^1 x 11^2

Step 2: Now we have the prime factors of 726. Since 726 is not a perfect square, we cannot pair all the factors evenly. Thus, prime factorization doesn't yield an exact square root for 726.

The long division method is used for non-perfect square numbers. Here's how to find the square root of 726 using this method:

Step 1: Group the digits of 726 from right to left. We group it as 26 and 7.

Step 2: Find n whose square is closest to 7. Use n = 2 because 2 x 2 = 4, which is less than 7. The quotient is 2, and after subtracting 4 from 7, the remainder is 3.

Step 3: Bring down 26, making the new dividend 326. Double the quotient (2), giving the new divisor as 4.

Step 4: Find n such that (4n) x n ≤ 326. Let's use n = 7, giving us 47 x 7 = 329, which is too large, so let's try n = 6.

Step 5: 46 x 6 = 276. Subtract 276 from 326, leaving a remainder of 50.

Step 6: Add a decimal point and bring down two zeros, making it 5000. Use 532 x 3 = 1596, which is too large.

Step 7: Adjust n to get a closer approximation. Continuously apply these steps until you reach the desired decimal precision.

The approximation method is an easier way to estimate square roots:

Step 1: Identify the nearest perfect squares around 726. The nearest smaller perfect square is 625 (25^2), and the nearest larger is 729 (27^2). Thus, √726 is between 25 and 27.

Step 2: Use the formula for approximation: (Given number - smaller perfect square) / (larger perfect square - smaller perfect square) Using (726 - 625) / (729 - 625) = 101 / 104 ≈ 0.971. Adding this to the base of 25: 25 + 0.971 = 25.971, so the square root of 726 is approximately 25.971.

• Square root: A square root is the inverse of squaring a number. For example, 5^2 = 25, so √25 = 5.

• Irrational number: An irrational number cannot be expressed as a simple fraction. For instance, √726 is irrational.

• Perfect square: A perfect square is a number that has an integer as its square root, such as 25, whose square root is 5.

• Prime factorization: Expressing a number as a product of prime numbers, e.g., 726 = 2 x 3 x 11 x 11.

• Long division method: A step-by-step technique to find the square root of non-perfect squares.