Square Root of 6800

The square root is the inverse of squaring a number. 6800 is not a perfect square. The square root of 6800 is expressed in both radical and exponential forms. In the radical form, it is expressed as √6800, whereas in the exponential form it is expressed as (6800)^(1/2). √6800 ≈ 82.462, which is an irrational number because it cannot be expressed in the form of p/q, where p and q are integers and q ≠ 0.

The prime factorization method is used for perfect square numbers. However, for non-perfect square numbers, the long-division method and approximation method are used. Let us now learn the following methods:

• Prime factorization method
• Long division method
• Approximation method

The product of prime factors is the prime factorization of a number. Now let us look at how 6800 is broken down into its prime factors.

Step 1: Finding the prime factors of 6800 Breaking it down, we get 2 × 2 × 2 × 2 × 5 × 5 × 17: 2^4 × 5^2 × 17

Step 2: Now that we found the prime factors of 6800, the second step is to make pairs of those prime factors. Since 6800 is not a perfect square, the digits of the number can’t be fully grouped into pairs.

Therefore, calculating √6800 using prime factorization directly is not possible.

The long division method is particularly used for non-perfect square numbers. In this method, we should check the closest perfect square number for the given number. Let us now learn how to find the square root using the long division method, step by step:

Step 1: To begin with, we need to group the numbers from right to left. In the case of 6800, we need to group it as 68 and 00.

Step 2: Now we need to find n whose square is less than or equal to 68. We can say n is 8 because 8 × 8 = 64, which is less than 68. Now the quotient is 8, and after subtracting 64 from 68, the remainder is 4.

Step 3: Bring down the next pair of numbers (00) to get a new dividend of 400.

Step 4: Double the quotient obtained so far (8), giving us 16, which will be part of our new divisor.

Step 5: Find a digit x such that 16x × x ≤ 400. The suitable digit is 2, because 162 × 2 = 324.

Step 6: Subtract 324 from 400, and the remainder is 76. The quotient is now 82.

Step 7: Since the dividend is less than the divisor, we need to add a decimal point. Adding the decimal point allows us to add two zeroes to the dividend. Now the new dividend is 7600.

Step 8: Find the new divisor, 164y × y ≤ 7600. The suitable digit is 4, as 1644 × 4 = 6576.

Step 9: Subtract 6576 from 7600 to get 1024.

Step 10: Continue this process until we reach the desired decimal places.

The approximate result for √6800 is 82.462.

The approximation method is another method for finding square roots. It is an easy method to find the square root of a given number. Now let us learn how to find the square root of 6800 using the approximation method.

Step 1: Find the closest perfect squares to √6800. The smallest perfect square less than 6800 is 6400 and the largest perfect square greater than 6800 is 7225. √6800 falls somewhere between 80 and 85.

Step 2: Use interpolation between the values of the closest perfect squares.

(6800 - 6400) / (7225 - 6400) = 400 / 825 ≈ 0.4848

Using this, estimate: 80 + 0.4848(5) = 82.424

The square root of 6800 is approximately 82.424.

• Square root: A square root is the inverse of squaring a number. Example: 4^2 = 16, and the inverse of the square is the square root, so √16 = 4.
   
• Irrational number: An irrational number cannot be written in the form of p/q, where q is not equal to zero and p and q are integers.
   
• Prime factorization: Prime factorization involves breaking down a number into a product of its prime factors. Example: 6800 = 2^4 × 5^2 × 17.
   
• Long division method: The long division method is a step-by-step approach for finding the square root of non-perfect squares.
   
• Principal square root: The principal square root is the non-negative square root of a number, typically used in practical applications.