Square Root of 899

The square root is the inverse of the square of the number. 899 is not a perfect square. The square root of 899 is expressed in both radical and exponential form. In the radical form, it is expressed as √899, whereas 899^(1/2) in the exponential form. √899 ≈ 29.9833, 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 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 899 can be broken down into its prime factors.

Step 1: Finding the prime factors of 899 Breaking it down, we get 29 x 31.

Step 2: Since 899 is not a perfect square, it cannot be grouped into pairs of equal prime factors.

Therefore, calculating √899 using prime factorization is not feasible.

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 899, we can group it as 89 and 9.

Step 2: We need to find a number n whose square is less than or equal to 89. Here, n is 9 because 9 x 9 = 81 is less than 89. Now the quotient is 9 and the remainder is 8.

Step 3: Bring down 9 to make the new dividend 89. Add the old divisor (9) with itself to get 18 as the new divisor.

Step 4: Determine n such that 18n x n ≤ 89. Using n = 4, we have 184 x 4 = 736.

Step 5: Subtract 736 from 899 to get a remainder of 163.

Step 6: Since the remainder is less than the divisor, we add a decimal point and two zeroes to the remainder, making it 16300.

Step 7: Find a new divisor that can divide 16300. This divisor is 9 because 1849 x 9 = 16641.

Step 8: Subtracting 16641 from 16300, we get -341, indicating an overestimate. Adjust the divisor and continue the process till you get a sufficient decimal approximation.

So the square root of √899 is approximately 29.9833.

The approximation method is another easy method for finding square roots of given numbers. Let us learn how to find the square root of 899 using the approximation method.

Step 1: Determine the closest perfect squares around 899.

The closest are 841 (29^2) and 900 (30^2).

Therefore, √899 falls between 29 and 30.

Step 2: Apply the formula:

(Given number - smallest perfect square) / (Greater perfect square - smallest perfect square).

Using the formula: (899 - 841) / (900 - 841) = 58 / 59 ≈ 0.9833.

Step 3: Add the integer part of the closest perfect square root to the decimal part: 29 + 0.9833 ≈ 29.9833.

So the square root of 899 is approximately 29.9833.

• 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, it is always the positive square root that has more prominence due to its uses in the real world. That is why it is known as the principal square root.
   
• Factors: Factors are numbers we can multiply together to get another number. For example, factors of 899 are 1, 29, 31, and 899.
   
• Long division method: The long division method is a way to divide numbers to find the quotient, and it can be used to determine the square root of non-perfect square numbers through a systematic approach.