Square Root of 869

The square root is the inverse of the square of the number. 869 is not a perfect square. The square root of 869 is expressed in both radical and exponential form. In radical form, it is expressed as √869, whereas in exponential form, it is expressed as (869)^(1/2). √869 ≈ 29.4788, 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 known as the prime factorization of a number. Now let us look at how 869 is broken down into its prime factors.

Step 1: Finding the prime factors of 869

Breaking it down, we get 869 = 11 x 79

Step 2: Now, we found out the prime factors of 869. The second step is to make pairs of those prime factors. Since 869 is not a perfect square, we cannot group the digits into pairs. Therefore, calculating 869 using prime factorization 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 869, we need to group it as 69 and 8.

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

Step 3: Now, let us bring down 69, which is the new dividend. Add the old divisor with the same number: 2 + 2 = 4, which will be our new divisor.

Step 4: The new divisor will be the sum of the dividend and quotient. Now we get 4n as the new divisor, and we need to find the value of n.

Step 5: The next step is finding 4n × n ≤ 469. Let us consider n as 1, now 41 x 1 = 41.

Step 6: Subtract 41 from 46, the difference is 5, and the quotient is 21.

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 500.

Step 8: Now, we need to find the new divisor that is 294 because 294 x 1 = 294.

Step 9: Subtracting 294 from 500, we get the result 206.

Step 10: Now, the quotient is 29.4.

Step 11: Continue doing these steps until we get two numbers after the decimal point. If there are no decimal values, continue until the remainder is zero.

So the square root of √869 ≈ 29.48

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 869 using the approximation method.

Step 1: Now, we have to find the closest perfect square of √869. The smallest perfect square less than 869 is 841, and the largest perfect square greater than 869 is 900. √869 falls somewhere between 29 and 30.

Step 2: Now, we need to apply the formula that is (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square) Using the formula (869 - 841) / (900 - 841) ≈ 0.478

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 29 + 0.478 = 29.478 Thus, the square root of 869 is approximately 29.478.

• Square root: A square root is the inverse of a square. Example: 4^2 = 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.
   
• Principal square root: A number has both positive and negative square roots; however, the positive square root has more prominence due to its uses in the real world. This is the reason it is also known as the principal square root.
   
• Prime factorization: Prime factorization is expressing a number as a product of its prime factors.
   
• Decimal point: A decimal point separates the whole number from the fractional part in a decimal number. For example, in 29.478, the decimal point separates 29 from 478.