Square Root of 772

The square root is the inverse of the square of a number. 772 is not a perfect square. The square root of 772 is expressed in both radical and exponential form. In the radical form, it is expressed as √772, whereas in exponential form, it is (772)^(1/2). √772 ≈ 27.78489, 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 772 is broken down into its prime factors.

Step 1: Finding the prime factors of 772 Breaking it down, we get 2 × 2 × 193: 2^2 × 193

Step 2: Since 772 is not a perfect square, the digits of the number can’t be grouped in pairs.

Therefore, calculating 772 using prime factorization alone cannot provide an exact square root.

The long division method is particularly used for non-perfect square numbers. Let's learn how to find the square root using the long division method, step by step.

Step 1: Group the digits of 772 from right to left. In this case, we have one group of two digits (72) and one group of one digit (7).

Step 2: Find the largest number whose square is less than or equal to 7. This number is 2, as 2^2 = 4. Subtract 4 from 7, leaving a remainder of 3.

Step 3: Bring down the next pair of digits (72) to get 372.

Step 4: Double the divisor (2) and find the next digit of the quotient such that (20 + n) × n is less than or equal to 372. The number is 7, as 27 × 7 = 189.

Step 5: Subtract 189 from 372 to get 183. Bring down two zeros to make it 18300.

Step 6: Repeat the process to find the next digit in the decimal places. Continue this process to find the square root to the desired precision.

Thus, √772 ≈ 27.78.

The approximation method is another method for finding square roots. Now let us learn how to approximate the square root of 772.

Step 1: Identify the perfect squares closest to 772.

The smallest perfect square less than 772 is 729, and the largest perfect square greater than 772 is 784. So, √772 falls between 27 and 28.

Step 2: Apply the formula: (Given number - smallest perfect square) / (largest perfect square - smallest perfect square).

Using this formula: (772 - 729) / (784 - 729) = 43 / 55 ≈ 0.7818. Adding the approximate decimal to the lower perfect square root gives 27 + 0.7818 ≈ 27.78.

• 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, which 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, the positive square root is often used in real-world applications. This is known as the principal square root.

• Prime factorization: Prime factorization is the process of expressing a number as a product of its prime factors. For example, the prime factorization of 18 is 2 × 3 × 3.

• Long division method: The long division method is a step-by-step process used to find the square root of a number, especially useful for non-perfect squares.