Square Root of 792

The square root is the inverse of squaring a number. 792 is not a perfect square. The square root of 792 can be expressed in both radical and exponential form. In radical form, it is expressed as √792, whereas in exponential form, it is expressed as (792)^(1/2). The value of √792 ≈ 28.14285, which is an irrational number because it cannot be expressed as a simple fraction.

The prime factorization method is used for perfect square numbers. However, for non-perfect squares like 792, methods such as the long division method and approximation method are utilized. Let us now explore these methods: 

• Prime factorization method
• Long division method
• Approximation method

The prime factorization of a number involves expressing it as a product of its prime factors. Let's see how 792 can be broken down into its prime factors.

Step 1: Finding the prime factors of 792 Breaking it down, we get 2 × 2 × 2 × 3 × 3 × 11: 2³ × 3² × 11

Step 2: Now that we have found the prime factors of 792, we attempt to make pairs of those prime factors. Since 792 is not a perfect square, the digits cannot all be grouped into pairs, making calculation of the exact square root using prime factorization impossible for simplification.

The long division method is particularly useful for non-perfect square numbers. Here’s how to find the square root using the long division method, step by step:

Step 1: Begin by grouping the digits of 792 starting from the right: 92 and 7.

Step 2: Find the largest number whose square is less than or equal to 7. Here, 2 × 2 = 4, which is less than 7. Write 2 as the first digit of the quotient.

Step 3: Subtract 4 from 7, leaving a remainder of 3. Bring down the next pair of digits, 92, to make 392.

Step 4: Double the divisor 2, getting 4, and determine a new digit n such that 4n × n ≤ 392. Choosing n = 7 gives 47 × 7 = 329.

Step 5: Subtract 329 from 392, resulting in a remainder of 63.

Step 6: Add a decimal point to the quotient and bring down two zeros, making the new dividend 6300.

Step 7: Double the current quotient (27), making it 54. Find a new digit n such that 54n × n ≤ 6300. Here, n = 1 gives 541 × 1 = 541.

Step 8: Subtract 541 from 6300, getting a remainder of 5759. Continue this process to get more decimal places until you reach the desired precision.

The approximate square root of 792 is 28.14.

The approximation method provides an easy way to estimate the square root of a number. Here’s how to approximate the square root of 792:

Step 1: Identify the nearest perfect squares around 792.

The smallest perfect square less than 792 is 784 (28²) and the largest perfect square greater than 792 is 841 (29²). Therefore, √792 lies between 28 and 29.

Step 2: Apply linear interpolation to approximate: (792 - 784) ÷ (841 - 784) = 8 ÷ 57 ≈ 0.14 Adding this to the lower bound of 28 gives 28 + 0.14 = 28.14.

Thus, the approximate square root of 792 is 28.14.

• Square root: The square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 9 is 3.

• Irrational number: An irrational number cannot be expressed as a simple fraction. The decimal goes on forever without repeating.

• Approximation: The process of finding a value that is close enough to the right answer, usually within some small error.

• Interpolation: A method of estimating values between two known values.

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