Square Root of 800

The square root is the inverse of the square of the number. 800 is not a perfect square. The square root of 800 is expressed in both radical and exponential form. In the radical form, it is expressed as √800, whereas in the exponential form it is (800)^(1/2). √800 ≈ 28.28427, 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, the prime factorization method is not used for non-perfect square numbers where 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 800 is broken down into its prime factors.

Step 1: Finding the prime factors of 800 Breaking it down, we get 2 x 2 x 2 x 2 x 2 x 5 x 5: 2^5 x 5^2

Step 2: Now we found out the prime factors of 800. The second step is to make pairs of those prime factors. Since 800 is not a perfect square, the digits of the number can’t be grouped into complete pairs.

Therefore, calculating 800 using prime factorization involves taking out pairs and simplifying under the radical.

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 800, we need to group it as 00 and 8.

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

Step 3: Now let us bring down 00, which is the new dividend, making it 400. Add the old divisor with the same number 2 + 2, we get 4, which will be our new divisor.

Step 4: The new divisor will be 4n, where n is a digit that when multiplied by 4n gives us a number less than or equal to 400. Let’s consider n as 7, now 47 x 7 = 329.

Step 5: Subtract 329 from 400, the difference is 71, and the quotient is 27.

Step 6: 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 7100.

Step 7: Now we need to find the new divisor that is 5 because 545 x 5 = 2725.

Step 8: Subtracting 2725 from 7100, we get the result 4375.

Step 9: Continue doing these steps until we get two numbers after the decimal point. Suppose if there is no decimal value, continue till the remainder is zero.

So the square root of √800 is approximately 28.28.

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 800 using the approximation method.

Step 1: Now we have to find the closest perfect squares of √800.

The smallest perfect square less than 800 is 784, and the largest perfect square greater than 800 is 841. √800 falls somewhere between 28 and 29.

Step 2: Now we need to apply the formula that is (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Going by the formula (800 - 784) ÷ (841 - 784) ≈ 0.28.

Using the formula, we identified the decimal point of our square root. The next step is adding the value we got initially to the decimal number, which is 28 + 0.28 ≈ 28.28, so the square root of 800 is approximately 28.28.

• Square root: A square root is the inverse of a square. Example: 4² = 16, and the inverse of the square is the square root, that is, √16 = 4.

• Irrational number: An irrational number is a number that cannot be written in the form of p/q, where q is not equal to zero and p and q are integers.

• Perfect square: A number that is the square of an integer. For example, 16 is a perfect square because it is 4².

• Long division method: A method used to find the square root of non-perfect square numbers by dividing and finding the quotient iteratively.

• Decimal: If a number has a whole number and a fraction in a single number, then it is called a decimal. For example, 7.86, 8.65, and 9.42 are decimals.