Square Root of 872

The square root is the inverse operation of squaring a number. 872 is not a perfect square. The square root of 872 can be expressed in both radical and exponential forms. In radical form, it is expressed as √872, whereas in exponential form, it is expressed as (872)^(1/2). Calculating the approximate value, √872 ≈ 29.5305, which is an irrational number because it cannot be expressed in the form p/q, where p and q are integers and q ≠ 0.

For perfect square numbers, the prime factorization method is often used. However, for non-perfect square numbers like 872, methods such as the long-division method and approximation method are more suitable. Let us explore these methods: 

• Prime factorization method 
• Long division method 
• Approximation method

Prime factorization involves expressing a number as a product of prime factors. However, for non-perfect squares like 872, we cannot proceed with pairing prime factors. Let us look at how 872 is broken down into its prime factors.

Step 1: Finding the prime factors of 872

Breaking it down, we get 2 x 2 x 2 x 109: 2^3 x 109

Step 2: Since 872 is not a perfect square, the digits of the number cannot be grouped in pairs. Thus, calculating √872 using prime factorization directly is not feasible.

The long division method is useful for finding the square root of non-perfect square numbers. Here's how you can find the square root using this method:

Step 1: Group the numbers from right to left in pairs. For 872, group as 87 and 2.

Step 2: Find a number whose square is ≤ 87. The number is 9 because 9^2 = 81 ≤ 87. The quotient becomes 9, and the remainder is 6 (87 - 81).

Step 3: Bring down the next pair, 2, making the new dividend 62.

Step 4: Double the quotient (9) to get 18, which will be part of the new divisor.

Step 5: Determine a digit, d, such that (180 + d) × d ≤ 620. The digit is 3, making the divisor 183.

Step 6: Multiply and subtract: 183 × 3 = 549, and 620 - 549 = 71.

Step 7: Add a decimal point and bring down pairs of zeros to continue the process until you reach the desired precision.

The square root of 872 using the long division method is approximately 29.5305.

The approximation method is another way to find the square root of a number. Here's how to find the square root of 872 using this method:

Step 1: Identify the closest perfect squares around 872. The smallest perfect square is 841 (29^2) and the largest is 900 (30^2). √872 is between 29 and 30.

Step 2: Use the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). (872 - 841) / (900 - 841) = 31 / 59 ≈ 0.525

Step 3: Add this to the smaller square root: 29 + 0.525 = 29.525

So, the square root of 872 is approximately 29.525.

• Square root: The square root is the inverse of squaring a number. Example: 5^2 = 25, and the square root of 25 is √25 = 5.
   
• Irrational number: An irrational number cannot be expressed as a simple fraction where both numerator and denominator are integers, with the denominator not being zero.
   
• Long division method: A technique used to find the square root of non-perfect-square numbers through systematic division.
   
• Radical expression: An expression that includes a root symbol, such as a square root or cube root.
   
• Perfect square: A number that is the square of an integer. Example: 16 is a perfect square because 4^2 = 16.