Square Root of 0.0008

The square root is the inverse operation of squaring a number. Since 0.0008 is not a perfect square, its square root can be expressed in radical and exponential forms. In radical form, it is √0.0008, and in exponential form, it is (0.0008)^(1/2). The square root of 0.0008 is approximately 0.028284, which is an irrational number because it cannot be expressed as a simple fraction.

For perfect squares, the prime factorization method is used, but for non-perfect squares like 0.0008, methods such as long division and approximation are more appropriate.

Let's explore these methods: 

• Long division method
• Approximation method

The long division method is useful for non-perfect squares. Here's how to find the square root of a number using this method:

Step 1: Start by pairing digits from right to left. For 0.0008, group as 08.

Step 2: Determine the largest number whose square is less than or equal to 8. Here, 2 works because 2² = 4. The quotient is 2, and the remainder is 8 - 4 = 4.

Step 3: Bring down two zeros, making the new dividend 400. Double the quotient (2), giving us 4.

Step 4: Find n such that (4n) × n ≤ 400. If n = 7, (47) × 7 = 329.

Step 5: Subtract 329 from 400, leaving 71.

Step 6: Add two zeros to make it 7100 and repeat the process until you achieve the desired precision.

The square root of 0.0008 is approximately 0.028284.

The approximation method provides a straightforward way to find square roots. Here's how to approximate the square root of 0.0008:

Step 1: Identify the perfect squares closest to 0.0008. The closest are 0.0004 (0.02²) and 0.0016 (0.04²). Thus, √0.0008 is between 0.02 and 0.04.

Step 2: Use interpolation: (Number - Lower perfect square) / (Upper perfect square - Lower perfect square). For example: (0.0008 - 0.0004) / (0.0016 - 0.0004) = 0.4 / 1.2 = 0.3333.

Step 3: Add this to the lower bound: 0.02 + 0.3333(0.02) = 0.026666.

Refining further yields an approximation of 0.028284.

• Square root: The square root is the inverse operation of squaring a number. For example, 4² = 16, so √16 = 4.
   
• Irrational number: A number that cannot be expressed as a simple fraction, such as the square root of a non-perfect square.
   
• Approximation: The process of finding a value close to the actual value. For example, √0.0008 ≈ 0.028284.
   
• Decimal: A numerical value with a fractional part, represented with a dot, such as 0.03, 0.25.
   
• Long division method: A technique used to find the square root of non-perfect squares through successive approximations.