Square Root of 709

The square root is the inverse of the square of the number. 709 is not a perfect square. The square root of 709 is expressed in both radical and exponential form. In radical form, it is expressed as √709, whereas in exponential form it is expressed as (709)^(1/2). √709 ≈ 26.62705, 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 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 709 is broken down into its prime factors.

Step 1: Finding the prime factors of 709 709 is a prime number, so it cannot be broken down further. The prime factorization of 709 is simply 709^1.

Step 2: Since 709 is not a perfect square, calculating √709 using prime factorization is not feasible. We will use other methods such as the long division method or approximation method.

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 709, we group it as 09 and 7.

Step 2: Now we need to find n whose square is ≤ 7. We can say n as ‘2’ because 2^2 = 4, which is less than 7. The quotient is 2, and after subtracting 4 from 7, we get a remainder of 3.

Step 3: Bring down 09, making the new dividend 309. Double the quotient (2) and write it as 4_, where _ is the next digit of the divisor. Step 4: Find a digit n such that 4n × n ≤ 309. Using trial, we find n = 6, as 46 × 6 = 276.

Step 5: Subtract 276 from 309, leaving a remainder of 33. The quotient so far is 26.

Step 6: Since the dividend is less than the divisor, we need to add a decimal point and bring down pairs of zeroes. The next dividend is 3300.

Step 7: Find a new divisor by doubling the current quotient (26) and adding the next digit. Repeat the process to find the next digits of the quotient until the desired precision is achieved.

Step 8: Continue until you get two numbers after the decimal point. The approximate square root of √709 is 26.63.

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 709 using the approximation method.

Step 1: Find the closest perfect squares surrounding 709. The smallest perfect square less than 709 is 676 (26^2), and the largest perfect square greater than 709 is 729 (27^2). √709 falls between 26 and 27.

Step 2: Use the formula: (Given number - smaller perfect square) / (Larger perfect square - smaller perfect square). (709 - 676) / (729 - 676) = 33 / 53 ≈ 0.6226 Using the formula, we identified the decimal point of our square root. Add this value to the smaller integer square root: 26 + 0.6226 ≈ 26.63. Therefore, the square root of 709 is approximately 26.63.

• Square root: A square root is the inverse operation of squaring a number. For example, 4^2 = 16, and the square root of 16 is √16 = 4.

• Irrational number: An irrational number cannot be expressed as a simple fraction. For example, √709 is irrational.

• Prime number: A prime number has only two distinct positive divisors: 1 and itself. For example, 709 is a prime number.

• Radical: A symbol (√) used to denote the root of a number. For example, √709 represents the square root of 709.

• Approximation: The process of finding a value that is close to the actual value. For example, √709 is approximately 26.63.