Square Root of 808

The square root is the inverse operation of squaring a number. 808 is not a perfect square. The square root of 808 can be expressed in both radical and exponential forms. In radical form, it is expressed as √808, whereas in exponential form it is expressed as (808)^(1/2). √808 ≈ 28.414, which is an irrational number because it cannot be expressed as a simple fraction p/q, where p and q are integers and q ≠ 0.

The prime factorization method is typically used for perfect squares. However, for non-perfect squares like 808, methods such as the long-division method and approximation method are used. Let's explore these methods:

• Prime factorization method
• Long division method
• Approximation method

Prime factorization involves breaking a number down into its prime factors. Now, let's see how 808 is broken down:

Step 1: Finding the prime factors of 808 Breaking it down, we get 2 x 2 x 2 x 101: 2^3 x 101^1

Step 2: Now that we have the prime factors of 808, we attempt to make pairs of those prime factors. Since 808 is not a perfect square, pairing is not possible.

Therefore, calculating √808 using prime factorization alone isn't feasible.

The long division method is useful for non-perfect square numbers. Here’s how to use it for finding the square root step by step:

Step 1: To begin with, group the digits in pairs from right to left. For 808, group it as 8 and 08.

Step 2: Find a number whose square is less than or equal to the first group (8). This number is 2, since 2^2 = 4. Subtract 4 from 8, leaving a remainder of 4.

Step 3: Bring down the next pair, 08, making the new dividend 408.

Step 4: Double the quotient obtained so far (2) to get 4. This becomes part of the new divisor.

Step 5: Find a digit n such that 4n × n is less than or equal to 408. The correct n is 6, since 46 × 6 = 276.

Step 6: Subtract 276 from 408, leaving a remainder of 132.

Step 7: Since the dividend is less than the divisor, add a decimal point and bring down 00, making it 13200.

Step 8: The new divisor becomes 566 (46 with an added digit 6), and the next digit in the quotient is 2. Thus, 562 × 2 = 1124.

Step 9: Subtract 1124 from 13200, resulting in 1176.

Step 10: Repeat these steps to refine the quotient. Eventually, the square root of 808 is approximately 28.41.

The approximation method is another way to estimate square roots. Here's how to approximate √808:

Step 1: Identify the closest perfect squares around 808. The nearest perfect squares are 784 (28^2) and 841 (29^2). So, √808 is between 28 and 29.

Step 2: Use the approximation formula: (Given number - smaller perfect square) / (larger perfect square - smaller perfect square). For 808, this is (808 - 784) / (841 - 784) = 24 / 57 ≈ 0.42.

Step 3: Add this to the smaller root: 28 + 0.42 = 28.42. So, the approximate square root of 808 is 28.42.

• Square root: The square root is an operation that finds a number which, when multiplied by itself, gives the original number. For example, √49 = 7.

• Irrational number: An irrational number is a number that cannot be expressed as a fraction of two integers. For example, √2 is irrational.

• Perfect square: A perfect square is a number that is the square of an integer. For example, 36 is a perfect square because it is 6^2.

• Long division method: A systematic method for finding square roots of non-perfect squares using division.

• Approximation method: A technique for estimating the square root of a number by finding nearby perfect squares and interpolating between them.