Square Root of 864

The square root is the inverse operation of squaring a number. 864 is not a perfect square. The square root of 864 can be expressed in both radical and exponential forms. In radical form, it is expressed as √864, whereas in exponential form, it is (864)^(1/2). The approximate value of √864 is 29.39388, which is an irrational number because it cannot be expressed as a fraction of two integers, p/q, where q ≠ 0.

For perfect squares, the prime factorization method is used. However, for non-perfect squares like 864, methods such as long division and approximation are more suitable. Let us 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 break down 864 into its prime factors:

Step 1: Finding the prime factors of 864

Breaking it down, we get 2 × 2 × 2 × 2 × 2 × 3 × 3 × 3: 2^5 × 3^3

Step 2: Pair the prime factors. Since 864 is not a perfect square, there will be unpaired factors. Therefore, calculating √864 using prime factorization directly is not feasible.

The long division method is particularly useful for non-perfect square numbers. This method involves finding the square root step by step:

Step 1: Group the numbers from right to left. For 864, group it as 64 and 8.

Step 2: Find n whose square is less than or equal to 8. Here, n = 2 since 2 × 2 = 4 ≤ 8. Subtract 4 from 8 to get a remainder of 4. Bring down 64 to form 464.

Step 3: Double the quotient (2 in this case) to get 4. The new divisor is 4x, and we need to find x such that 4x × x ≤ 464.

Step 4: Choose x = 7, since 47 × 7 = 329. Subtract 329 from 464 to get 135.

Step 5: Add a decimal point and bring down 00 to make it 13500.

Step 6: Continue the process to get a more precise approximation, continuing until you reach the desired accuracy.

Thus, the square root of 864 is approximately 29.39.

The approximation method is an easy way to estimate square roots. Here's how to find the square root of 864 using this method:

Step 1: Identify the perfect squares closest to 864. The smaller perfect square is 841 (√841 = 29) and the larger is 900 (√900 = 30). So √864 lies between 29 and 30.

Step 2: Use the formula to approximate: (Given number - smaller perfect square) ÷ (Larger perfect square - smaller perfect square) (864 - 841) ÷ (900 - 841) = 23 ÷ 59 ≈ 0.39

Step 3: Add this decimal to the smaller root: 29 + 0.39 = 29.39

Therefore, the approximate square root of 864 is 29.39.

• Square root: The square root of a number is a value that, when multiplied by itself, gives the original number. Example: 4^2 = 16, and the square root of 16 is √16 = 4.
   
• Irrational number: A number that cannot be expressed as a simple fraction, with non-repeating decimal expansions.
   
• Perfect Square: A number that can be expressed as the square of an integer. For example, 16 is a perfect square because it is 4 × 4.
   
• Prime Factorization: Expressing a number as the product of its prime factors. Example: the prime factorization of 864 is 2^5 × 3^3.
   
• Long Division Method: A technique used to find the square root of non-perfect squares by dividing the number into groups and calculating iteratively.